# Two-Scale Network Dynamic Model for Energy Commodity Processes

Authors: Olusegun M. Otunuga; Gangaram Ladde
Journal: Journal of Energy
 (2020)

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## Abstract

In this work, we examine the relationship between different energy commodity spot prices. To do this, multivariate stochastic models with and without external random interventions describing the price of energy commodities are developed. Random intervention process is described by a continuous jump process. The developed mathematical model is utilized to examine the relationship between energy commodity prices. The time-varying parameters in the stochastic model are estimated using the recently developed parameter identification technique called local lagged adapted generalized method of moment (LLGMM). The LLGMM method provides an iterative scheme for updating statistic coefficients in a system of generalized method of moment/observation equations. The usefulness of the LLGMM approach is illustrated by applying to energy commodity data sets for state and parameter estimation problems. Moreover, the forecasting and confidence interval problems are also investigated (U.S. Patent Pending for the LLGMM method described in this manuscript).


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## Body

## 1. Introduction

Understanding the economy evolution in response to structural changes in the energy commodity network system is important to professional economists. The relationship between the different energy sources and their uses provide insights into many important energy issues. The quantitative behavior of energy commodities in which the trend in price of one commodity coincides with the trend in prices of other commodities has always raised the question of whether there is any relationship between prices of energy commodities. If there is any relationship, then what comes to mind is the extent to which one commodity influences the other. Petroleum, natural gas, coal, nuclear fuel, and renewable energy are termed as primary energy components of the energy goods network system because other sources of energy depend on them. Natural gas is usually found near petroleum. Hence, natural gas and crude oil are rivals in production and substitutes in consumption, and energy theory suggests that the two prices should be related. The electric power sector uses primary energy such as coal to generate electricity, which makes electricity a secondary rather than a primary energy source. According to the US Energy Information Administration (EIA), the major energy goods consumed in the United States are petroleum (oil), natural gas, coal, nuclear, and renewable energy. The majority of users are residential and commercial buildings, industry, transportation, and electric power generators. The pattern of fuel usage varies widely by sector [1]. For example, 71% of total petroleum oil provides 93% of the energy used for transportation; 23% of total petroleum oil provides 17% of energy used for residential and commercial use; 5% of total petroleum oil provides 40% of energy used for industrial use; but only 1% of total petroleum oil provides about 1% of the energy used to generate electric power, whereas coal provides 46% of the energy used to generate electric power and natural gas provides 20% of the energy used to generate electric power. This analysis suggests that the strength of interactions between coal and electricity will be stronger than that of crude oil and electricity, or natural gas and electricity.Energy price forecasts are highly uncertain. We might expect the price of the natural gas and crude oil to follow the same trend because they are often found mixed with oil in oil wells and also of the fact that natural gas is often used in petroleum refining and exploration. Recently, Ramberg and Parsons [2] showed that the cointegration relationship between natural gas and crude oil does not appear to be stable through time. They claimed that though there is cointegration between the two energy prices, there are also recurrent exogeneous factors such as seasonality, episodic heat waves, cold waves, and supply interruption from the hurricane affecting the trends in the prices. Brown and Yücel [3] also observed that the price of natural gas pulled away from oil prices in 2000, 2002, 2003, and late 2005. Oil prices are influenced by several factors, including some that have mainly short-term impacts and other factors, such as expectations about the future world demand for petroleum, other liquids, and production decisions of the Organization of the Petroleum Exporting Countries (OPEC) [1]. Supply and demand in the world oil market are balanced through responses to price movement with considerable complexity in the evolution of underlying supply and demand expectation process. For petroleum and other liquids, the key determinants of long-term supply and prices can be summarized in four broad categories [1]: the economics of non-OPEC supply, OPEC investment and production decisions, the economics of other liquids supply, and world demand for petroleum and other liquids.An understanding of how changes in the price of one energy commodity are expressed in terms of other energy commodity is needed. This would prove to be useful in predicting price behavior over the long run and further facilitates profit-maximizing processes. To check if there is really indeed a relationship between energy commodities, the need to be able to create a model which explains such commodity price relationship over short- and long-time intervals arises. The relationships between energy commodities have been addressed in [2–15]. The error correction model [2–5, 7, 8] is the most commonly used model by authors to describe the relationship between energy commodities. Moreover, Hartley et al. [7] have concluded that the U. S. natural gas and crude oil remain linked in their long-term movements. In addition, it is exhibited that there is strong evidence of a stable relationship between these two energy commodities which are affected by short-run seasonal fluctuations and other factors. The rule of thumb [7] has long been used in the energy industry to relate the natural gas prices to crude oil prices. The rule denoted by the 10-to-1 rule states that the price of natural gas is one-tenth of the price of crude oil prices. Similarly, the 6-to-1 rule states that the price of natural gas is one-sixth of the price of crude oil. It has been examined by Brown and Yücel [3] that these two rules do not perform well when used to assess the relationship between US natural gas price and West Texas Intermediate (WTI) crude oil price for the past 20+ years. Moreover, their error-correcting model specifies the relationship between the two commodities. Using this model, they show that when certain factors are taken into account, movements in crude oil prices can shape the price of natural gas. Vezzoli [9] in his work applies an ordinary least squares (OLS) regression on the log of natural gas and crude oil prices. Using this model, he was able to conclude that there is a relationship between natural gas and crude oil prices. Bachmeir and Griffin [4] showed that crude oil, coal, and natural gas in the United States have weak cross-cointegration using the error correction model. Ramberg and Parsons [2] show that any simple formula between natural gas and crude oil prices will leave a portion of the natural gas price unexplained. Furthermore, the relationship between natural gas and crude oil using a vector error correction model [2, 3] under the cointegration between the two energy commodities and other factors such as recurrent exogeneous factors are presented. Villar and Joutz [10] list some economic factors linking natural gas and crude oil prices, while testing for market integration in the United Kingdom during the time when natural gas was deregulated. Asche et al. [6] have integrated the prices of the energy commodities: natural gas, electricity, and crude oil.Most of this work is centered around the relationship between prices of energy commodities. We are interested in an interdependence of certain energy commodities. Moreover, a nonlinear multivariate interconnected stochastic model of energy commodities and sources with and without external random intervention processes is developed. Also, parameter and state estimation problems of continuous time nonlinear stochastic dynamic process are motivated to initiate an alternative innovative approach. This led to the introduction of the concept of statistic processes, namely, local sample mean and sample variance which further led to the development of an interconnected discrete-time dynamic system of local statistic processes and its mathematical model discussed in Otunuga et al. [16, 17]. This paved the way for extending the LLGMM [16] to a multivariate local lagged adapted generalized method of moments case. The parameters in the multivariate model are estimated using the LLGMM method. These estimated parameters help in analyzing the short- and long-term relationships between the energy commodities. It has been shown in Appendices A, B, C, and D of the work of Otunuga et al. [16] that the LLGMM parameter estimation scheme performs better than existing nonparametric statistical methods. A performance comparison (with appropriate statistical indices) of the LLGMM method with existing orthogonality condition based on generalized method of moments (OCBGMM) and aggregated generalized method of moments (AGMM) methods using the energy commodity stochastic model is discussed in their work. The method is applied to estimate time-varying parameter estimates in a stochastic differential equation governing energy commodities, stock price processes, and transmission of infectious diseases in the work of Otunuga et al. [16], Otunuga [18], and Mummert and Otunuga [19], respectively. Algorithm for implementing the LLGMM approach is presented in Otunuga et al. [17]. In this work, the usefulness of this approach is further illustrated by applying the technique to study the relationship between three energy commodity data sets: Henry Hub natural gas, crude oil, and coal data sets for state and parameter estimation problems. Interested readers are directed to the work of Otunuga et al. [16–18, 20] to read more about the LLGMM parameter estimation procedure. Moreover, the forecasting and confidence interval problems are also investigated.The organization of this paper is as follows.In Section2, we derive a multivariate stochastic dynamical model for energy commodities. In Section 3, the multivariate model derived in Section 2 is validated. Due to random intervention, we introduce interventions described by a continuous jump process in our model in Section 4. In Section 5, we discuss the discrete-time dynamic model for sample mean and covariance processes with jump using the work of Otunuga et al. in [16, 17]. In Section 6, we discuss about parametric estimation techniques. We construct an observation system from a nonlinear stochastic functional differential equation. In Section 7, using the method of moments [21], in the context of lagged adaptive expectation process [22], we briefly outline a procedure to estimate the state parameters for the dynamical model with jump and the model without jump locally. Moreover, the usefulness of computational algorithm is illustrated by applying the procedure to test for the relationship between Henry Hub natural gas, crude oil, and coal for the state and parameter estimation problems. In Section 8, the forecasting and confidence interval problems are also addressed.


## 2. Model Derivation

Letp=p1,p2,⋯,pnΤ be a vector of n energy commodity prices considered to have long-run or short-run relationship with each other, with pjt being the price of the jth energy commodity at time t. The economic principles of demand and supply processes suggest that the price of an energy commodity will remain within a given finite expected lower and upper bounds. We define uj∈R+=0,∞ and lj≥0 as the expected upper and lower limits of the jth energy commodity spot prices, respectively. In the absence of interactions between the energy commodities pj, j∈I1,n, where
(1)Ia,b=z∈ℤ:a≤z≤b,the market potential for the jth commodity per unit of time at time t can be characterized by uj−pjlj+pj. This simple idea leads to the following economic principle regarding the dynamic of the price pj of the jth energy commodity. The change in spot price of the jth energy commodity Δpjt=pjt+Δt−pjt over the interval of length ∣Δt∣ is directly proportional to the market potential price, that is,
(2)Δpjt∝uj−pjlj+pjΔt.This implies that(3)dpj=αjuj−pjlj+pjdt,for some constant αj. From this deterministic mathematical model, if αj>0, we note that as the price falls below the expected price uj, the price of the jth commodity rises, and as the price lies above uj, there is a tendency for the price to fall. Similar argument follows if αj<0. Hence uj is the equilibrium state of (3).In a real world situation, the expected upper price limituj of the jth commodity is not a constant parameter. It varies with time, and moreover it is subjected to random environmental perturbations. Therefore, we consider
(4)uj=yj+ξj,where ξj is a white noise process that characterizes the measure of random fluctuation of the upper price limit of the jth commodity; here yj stands for the mean of the energy spot price process of the jth commodity at time t. It is further assumed that yj is governed by a similar dynamic forces described in (3), that is,
(5)dyj=μjuj−yjυj+yjdt,where μj∈R+ is defined as the mean reversion rate of the mean of the jth commodity. By following the argument used in (4), we incorporate the effects of random environmental perturbations into the lower limit υj of the mean of the jth commodity:
(6)υj=vj+ej,where vj≥0, and ej is a white noise process describing the measure of random influence on the mean price of the jth commodity.Substituting expressions in (4) and (6) into (3) and (5), respectively, we obtain
(7)dyj=μjuj−yjvj+yjdt+μjuj−yjejtdtdpj=αjyj−pjlj+pjdt+αjlj+pjξjtdt.In the absence of interactions and using (7), the system of stochastic model for isolated expected spot and spot prices processes are described by the following nonlinear system of stochastic differential equations:
(8)dyj=μjuj−yjvj+yjdt+δj,juj−yjdWj,jt,yjt0=yj0,dpj=αjyj−pjlj+pjdt+σj,jlj+pjdZj,jt,pjt0=pj0,j∈I1,n,where
(9)μjejtdt=δj,jdWj,jt,αjξjtdt=σj,jdZj,jt,and δj,j,σi,j are nonnegative for j∈I1,n.In the presence of interactions, for eachj∈I1,n, we consider both deterministic and stochastic interaction functions. For each j∈I1,n, we define the jth aggregate interaction functions gj:t0,∞×Rn⟶R and hj:t0,∞×Rn⟶R for the jth mean energy spot price process yjt and the energy spot price process pjt in energy commodity market network system, respectively. Moreover, we assume that these functions have the following structural forms:
(10)gjt,y=gjt,kj,1y1,kj,2y2,⋯,kj,nyn,hjt,p=hjt,γj,1p1,γj,2p2,⋯,γj,npn,where kj,i and γj,i are elements of n×n interconnection matrices denoted by Eg and Eh, respectively. In (10), kj,i and γj,i represent a degree of interaction of the jth commodity with the ith commodity in the energy commodity market network system.For the matrixEg, kj,i=0 with fixed i∈I1,n if the ith commodity in the energy market network system does not influence the jth commodity. Likewise, for the matrix Eh, γj,i=0 with fixed i∈I1,n, if the jth commodity in the energy market network system subcomponent of p is totally unaffected by the influence of the ith commodity. Finally, we introduce interactions in the diffusion coefficients with respect to the jth commodity of the energy market network system under random environmental perturbations as ψj:t0,∞×Rn⟶Rn and Λj:t0,∞×Rn⟶Rn for each j∈I1,n. The diffusion part is of the form
(11)ψjt,y·ejtdt=∑l=1nψj,lt,yldWj,lt,Λjt,p·ξjtdt=∑l=1nΛj,lt,pldZj,lt,where ej and ξj are n-dimensional white noise processes; · stands for dot product.We assume that the interaction functions (10) and (11) have the following forms:
(12)gt,y=γt,yGt,y,ht,p=λt,pHt,p,ψt,y=γt,yΨt,y,Λt,p=λt,pΦt,p,where gt,y=g1t,y,⋯,gjt,y,⋯,gnt,y⊤, ht,p=h1t,p,⋯,hjt,p,⋯,hnt,p⊤ are defined in (10), ψt,y=ψj,lt,yn×n, and Λt,p=Λj,lt,pn×n, γt,y=diagu1−y1,⋯,uj−yj,⋯,un−yn and λt,p=diagl1+p1,⋯,lj+pj,⋯,ln+pn; G, and H are n×1 column vectors; Ψ=diagψ1,⋯,ψj,⋯,ψn and Φ=diagΛ1,⋯,Λj,⋯,Λn are block diagonal matrices; ψj=ψj,1,⋯,ψj,l,⋯,ψj,n, Λj=Λj,1,⋯,Λj,l,⋯,Λj,n. We also assume that G, H, Ψ, and Φ satisfy the local Lipschitz condition. This assumption implies that g, h, ψ, and Λ also satisfy local Lipschitz condition.Thus, the interconnected energy commodity network system is described by(13)dyj=uj−yjμjvj+yj+Gjt,ydt+δj,jdWj,jt+∑l=1nΨj,lt,ydWj,lt,yjt0=yj0,dpj=lj+pjαjyj−pj+Hjt,pdt+σj,jdZj,jt+∑l=1nΦj,lt,pdZj,lt,pjt0=pj0,j∈I1,n,where the parameters μj>0; αj>0; uj>0; vj≥0; lj≥0; δj,j>0; σj,j>0; and for j≠l, δj,l≥0; σj,l≥0; j,l∈I1,n; for j∈I1,n, Wj and Zj are n-dimensional independent Wiener processes defined on a filtered probability space (Ω,F,Ftt≥0,P); for l≠i, EdWj,ldWk,i=0, and for l=i, EdWj,ldWk,i=dt; the filtration function Ftt≥0 is right continuous; for each t≥0, each Ft contains all P-null sets in F; the n-dimensional random vectors yt0 and pt0 are Ft0 measurable.The network system of stochastic differential equations in (13) can be written as follows:
(14)dy=at,ydt+Yt,ydWt,yt0=y0,dp=bt,y,pdt+σt,pdZt,pt0=p0,where
(15)at,y=u1−y1μ1v1+y1+G1t,yu2−y2μ2v2+y2+G2t,y⋮un−ynμnvn+yn+Gnt,y,bt,y,p=l1+p1α1y1−p1+H1t,pl2+p2α2y2−p2+H2t,p⋮ln+pnαnyn−pn+Hnt,p,Yt,y=diagA1y,⋯,Ajy,..,Any,σt,p=diagB1p,⋯,Bjp,⋯,Bnp,and
(16)Ajy=uj−yjΨj,1Ψj,2⋯Ψj,j−1δj,j+Ψj,jΨj,j+1⋯Ψj,n,Bjp=lj+pjΦj,1Φj,2⋯Φj,j−1σj,j+Φj,jΦj,j+1⋯Φj,n;W=W1,⋯Wj,⋯,WnΤ and Z=Z1,⋯Zj,⋯,ZnΤ are block matrices; Wj=Wj,1,⋯Wj,2,⋯,Wj,nΤ, Zj=Zj,1,⋯Zj,2,⋯,Zj,nΤ; and Yt,y, σt,p are a n×n block matrix with each entries having order 1×n.In the next section, we outline the model validation problems of (14), namely, the existence and uniqueness of solution process.


## 3. Mathematical Model Validation

Here, we validate the mathematical model derived in Section2. We note that the classical existence and uniqueness theorem [23–25] is not directly applicable to (14). We need to modify the existence and uniqueness results. The modification is based on Theorem 3.4 [23]. We show the global existence of the solution process of a system of differential equations (14). We note that the rate functions a, b, Y, and σ in stochastic system of differential equations (14) do not satisfy the classical existence and uniqueness conditions [23]. However, these rate functions do satisfy the local Lipschitz condition. Therefore, we construct sequences of functions for the drift and diffusion coefficients of interconnected dynamic system (14) so that the classical conditions for existence and uniqueness theorem are applicable. The construction of modification scheme is as follows: First, we define a cylindrical subset t0,∞×Um of 0,∞×Rn for t0∈0,∞, m≥1, where Um is an n- dimensional sphere with radius m defined by
(17)Um=Bx0,m=x∈Rn:x−x0<m,for any m≥1. We note that Um is inscribed in n-dimensional parallelepiped ℝx−x0,m=−m,m×⋯×−m,m in Rn.The developed stochastic network model (14) can be written as:
(18)dy=amt,ydt+Ymt,ydWt,yt0=y0,dp=bmt,y,pdt+σmt,pdZt,pt0=p0,where
(19)amt,y=at,qy,m,Ymt,y=Yt,qy,m,bmt,y,p=bt,qy,m,qp,m,σmt,p=σt,qp,m.Here, for eachj∈I1,n and x∈Rn, we define qjx,m=max−m,minxj−x0j,m. Hence, qx,m=q1x,m,⋯,qjx,m,⋯,qnx,mΤ.Remark 1.
We observe thatqx,m satisfies the global Lipschitz condition on Rn with a Lipschitz constant 1. Using this, together with the local Lipschitz condition assumption on the drift and diffusion coefficients of stochastic differential equations (14), the modified rate coefficient functions in (18) satisfy the classical existence and uniqueness conditions [26, 27]. Thus, its solution is denoted by (ym,pm), for m≥1. Moreover, (y,p) is nonnegative whenever y0, p0∈R+n.Now, we apply Theorems 3.4 and 3.5 of [26] in the context of the modified system of stochastic differential equations (18) and Remark 1 to establish the global existence of solution of stochastic differential equations in (13). For this purpose, we outline the argument used in the proof of these theorems. In addition to the local Lipschitz conditions on the drift and diffusion coefficients, we further impose the following hypothesis on the coefficients:(H1)
(20)gjt,y≤a1,j+κjy,hjt,p≤a1,j′+γjp,∣ψj,lt,y∣≤a2,j+δ~j,ly,Λj,lt,p≤a2,j′+σ~j,lp,where for i∈I1,2, ai,j, ai,j′ are nonnegative; κj, γj, δ~j,l,σ~j,l∈R+. From (18), we further remark that the dynamic of the mean spot price vector y is decoupled with the dynamic of spot price p. Now, we first apply Theorems 3.4 and 3.5 of [23] in the context of modified system of stochastic differential equations (18) and hypothesis (H1) to establish the global existence of solution of the completely decoupled subsystem of stochastic differential equations in (18). For this purpose, we outline the argument used in the proof of these theorems.Definition 2.
Letτm be the first exit time of the solution process ym from the set By0,m. Define τ to be the (finite or infinite) limit of the monotone increasing sequence τm as m⟶∞.
(21)τ=limm→∞τm.We wish to show that(22)Probτ=∞=1.In the following, we present a result that is parallel to Theorem 3.5 [26] in the context of the completely decoupled subsystem of stochastic differential equation (14). For this purpose, it is enough to exhibit the global existence result for the transformed system (18).Lemma 3.
Form≥1, and y0∈R+n, let ymt=ymt,t0,y0 be the solution of the completely decoupled subsystem of (18), and let the hypothesis (H1) be satisfied. Let V1 be a function defined on t0,∞×R+n into R+ defined by
(23)V1t,y=lny2+e,
Then, there exists some constantc1>0 such that
(24)LV1≤c1V1,V1,m=infy>mV1t,y⟶∞asm⟶∞,where L is the differential operator with respect to (14); e=exp1.Moreover, the global existence of solution of the completely decoupled subsystem of (14) is follows.Proof.
It is obvious thatV1∈C1,2 on t0,∞×R+n→R+. In fact, ∂V1t,y/∂yj=2yj/y+e, ∂2V1t,y/∂yj2=2/y+e−4yj2/y2+e2, and ∂2V1t,y/∂yiyj=−4yiyj/y2+e2 exist and are continuous functions defined on t0,∞×R+n⟶R. Moreover, the L operator with respect to the completely decoupled component is as follows:
(25)LV1t,y=∑j=1nμjuj−yjvj+yj+gjt,y∂V1t,y∂yj+12∑j=1nδj,juj−yj+ψjjt,y2+∑l≠jnψj,l2t,y∂2V1t,y∂yj2+12∑i=1n∑j=1j≠in∑l≠i,jnψi,lt,yψj,lt,y+2δi,iui−yi+ψi,iψj,i∂2V1t,y∂yiyj=∑j=1nμj−yj−uj−vj22+uj+vj222yjy2+e+∑j=1n2gjt,yyjy2+e+12∑j=1nδj,juj−yj+ψj,j22y2+e−4yj2y2+e2+12∑j=1n∑l≠jnψj,l2t,y2y2+e−4yj2y2+e2−12∑i=1n∑j=1j≠in∑l≠i,jnψi,lt,ylψj,lt,y+2δi,iui−yi+ψi,iψj,i4yiyjy2+e2≤2∑j=1nμjuj+vj22yjy2+e+∑j=1n2gjt,yyjy2+e+∑j=1nδj,juj−yj+ψjjt,y2y2+e+∑j=1n∑l≠jnψj,l2t,yy2+e−12∑i=1n∑j=1j≠in∑l≠i,jnψi,lt,ylψj,lt,y+2δi,iui−yi+ψi,iψj,i4yiyjy2+e2≤2∑j=1nμjuj+vj22+∑j=1na1,j+κjy22+yj2y2+e+2∑j=1nδj,j2uj+12+a2,j+δ~j,j2+∑j=1n∑l≠jna2,j+δ~j,l2+2∑i=1n∑j≠in∑l≠i,jna2,i+δ~i,la2,j+δ~j,l+4∑i=1n∑j≠ina2,j+δ~j,iδi,iui+1+a2,i+δ~i,i≤c1V1t,y,where
(26)c1=2∑j=1nμjuj+vj22+1+∑j=1na1,j+κj2+2∑j=1nδj,j2uj+12+a2,j+δ~j,j2+∑j=1n∑l≠jna2,j+δ~j,l2+2∑i=1n∑j≠in∑l≠i,jna2,i+δ~i,la2,j+δ~j,l+4∑i=1n∑j≠ina2,j+δ~j,iδi,iui+1+a2,i+δ~i,i.
Furthermore,infy>mV1t,y⟶∞ as m⟶∞.
To show thatProbτ=∞=1, we define a function
(27)Vt,y=V1t,yexp−c1t−t0.
It is obvious thatLV≤0. By defining τmt=minτm,t; Yt=ymt for t<τm; and imitating the argument of Lemma 4 [23], we have
(28)EV1τmt,Yτmt≤ec1t−t0EV1t0,yt0.
Hence,(29)Probτm≤t≤ec1t−t0EV1t0,Yt0infy>m,u>t0V1u,y⟶0asm⟶∞by24.
The global existence and uniqueness of the solution of the first component of (18) follows by letting m⟶∞. Hence, from this and the fact that the solution process of transformed system (18) is almost surely identical with the solution process of the original system (14), we conclude the global existence and uniqueness of (14).
Following the idea of Lemma3, we present a global existence and uniqueness of solution of the system of stochastic differential equations governing the subsystem p in (14).Lemma 4.
Form≥1, and y0, p0∈R+n, let pmt=pmt,t0,p0 be the solution of the system of stochastic differential equations governing the subsystem p described in (18), and let the hypothesis (H1) be satisfied. Let V2 be a nonnegative function on t0,∞×R+n into R+ defined by
(30)V2t,p=lnp2+e+∑j=1nαj2∫tΤyjs+lj2ds,
Then, there exist a constantc>0 such that
(31)LV2≤cV2V2,m=infp>mV2t,p⟶∞asm⟶∞.where L is the differential operator with respect to (14); e=exp1.Moreover, the global existence of the solution of the system of stochastic differential equations governing the subsystemp described in (14) is as follows.Proof.
It is obvious thatV2∈C1,2 on t0,∞×R+n⟶R+. In fact, ∂V2t,p/∂t=−∑j=1nαj/2yjt+lj2, ∂V2t,p/∂pj=2pj/p2+e, ∂2V2t,p/∂pj2=2/p2+e−4pj2/p2+e2, and ∂2V2t,p/∂pipj=−4pipj/p2+e2 exist and are continuous functions defined on t0,∞×R+n⟶R. Moreover, the L operator with respect to the system of stochastic differential equations governing the subsystem p described in (14) is obtained as follows:
(32)LV2t,p=−∑j=1nαj2yjt+lj2+∑j=1nαjyj−pjlj+pj+hjt,p2pjp2+e,+12∑j=1nσj,jlj+pj+Λj,jt,p2+∑l≠jnΛj,lt,p22p2+e−4pj2p2+e2+12∑i=1n∑j=1j≠in∑l=1nΛi,lΛj,l+2σi,ili+pi+Λi,iΛj,i−4pipjp2+e2=−∑j=1nαj2yj+lj2+∑j=1nαj−pj−yj−lj22+yj+lj222pjp2+e+∑j=1n2hjt,ppjp2+e+12∑j=1nσj,jlj+pj+Λj,jt,p22p2+e−4pj2p2+e2+12∑j=1n∑l≠jnΛj,lt,p22p2+e−4pj2p2+e2−12∑i=1n∑j=1j≠in∑l=1nΛi,lΛj,l+2σi,ili+pi+Λi,iΛj,i4pipjp2+e2≤−∑j=1nαj2yj+lj2+∑j=1nαj2yj+lj2pjp2+e+∑j=1na1,j′+γjp2+pj2p2+e+∑j=1nσj,jlj+pj+Λj,jt,p2p2+e+∑j=1n∑l≠jna2,j′+σ~j,l2+2∑i=1n∑j≠in∑l≠i,jna2,i′+σ~i,la2,j′+σ~j,l+4∑i=1n∑j≠ina2,j′+σ~j,iσi,ili+1+a2,i′+σ~i,i≤cV2t,p,where
(33)c=1+∑j=1na1,j′+γj2+2∑j=1nσj,j2lj+12+a2,j′+σ~j,j2+∑j=1n∑l≠jna2,j′+σ~j,l2+2∑i=1n∑j≠in∑l≠i,jna2,i′+σ~i,la2,j′+σ~j,l+4∑i=1n∑j≠ina2,j′+σ~j,iσi,ili+1+a2,i′+σ~i,i.
Furthermore,infp>mV2t,p⟶∞ as m⟶∞. Following the final argument used in proving the global existence of y in Lemma 3, we conclude that there exists a unique global solution to the interconnected system of stochastic differential (14).In the next section, we discuss the case where we incorporate jump process into the system (y,p).


## 4. Energy Commodity Model with and without Jumps

Due to random interventions that affects the price of energy commodities, we introduce random perturbations described by a continuous jump in our model. We follow the approach discussed in [28, 29] where a class of stochastic hybrid dynamic processes is investigated.LetK≥0 be the number of jumps on the interval t0,T, for T>0. For K≠0, let T1,⋯,TK be the jump times over a time interval t0,T such that t0=T0≤T1<⋯<TK≤T, where Ti denotes the time at which the ith jump occurred in the system y,p. For K=0, no jump has occurred on the interval t0,T. We denote the ith subinterval by Ti−1≤t<Ti. Knowing the global existence and uniqueness solution process of the system (14) on the interval t0,T, T>0 in Section 3, for i∈I1,K∗, where K∗=K if TK=T, and K∗=K+1 if TK<T and K≠0, we consider the solution process on each subinterval Ti−1,Ti, between jumps. For i∈I1,K∗ and K=0, we have I1,K=∅ and I1,K∗=1. In this case, we consider the solution process on t0,T. The interconnected system is governed by the following system of continuous time stochastic process:
(34)dyi−1=ai−1t,ydt+Yi−1t,ydWt,yTi−1=yi−1,t∈Ti−1,Tidpi−1=bi−1t,y,pdt+σi−1t,pdZt,pTi−1=pi−1,t∈Ti−1,Ti,i∈I1,K∗yi=Πiyi−1Ti−,Ti−1,yi−1,pi=Θipi−1Ti−,Ti−1,yi−1,pi−1,where
(35)Πi=diagπ1i,π2i,⋯.,πni,Θi=diagθ1i,θ2i,⋯.,θni,yi−1t,Ti−1,yi−1,pi−1t,Ti−1,yi−1,pi−1 is the solution of the system of equation (34); for K≠0 and i∈I1,K∗, Πi and Θi are jump coefficient matrices. These jump coefficients are estimated by π^ji=yjTi/limt→Ti−yji−1t,Ti−1,yi−1; θ^ji=pjTi/limt→Ti−pji−1t,Ti−1,yi−1,pi−1 for j∈I1,n. Following the approach in [28, 29], the solution of (34) takes the following representation:
(36)yt,t0,y0=y0t,t0,y0,yt0=y0,t0≤t<T1,y1t,T1,y1,y1=yT1,T1≤t<T2,⋯yi−1t,Ti−1,yi−1,yi−1=yTi−1,Ti−1≤t<Ti,⋯yKt,TK,yK,yK=yTK,TK≤t≤T,pt,t0,y0,p0=p0t,t0,y0,p0,pt0=p0,t0≤t<T1,p1t,T1,y1,p1,p1=pT1,T1≤t<T2,⋯pi−1t,Ti−1,yi−1,pi−1,pi−1=pTi−1,Ti−1≤t<Ti,⋯pKt,TK,yK,pK,pK=pTK,TK≤t≤T,Remark 5.
In case of no jump, that isK=0, (34) and (36) reduce to
(37)dy=at,ydt+Yt,ydWt,yt0=y0,dp=bt,y,pdt+σt,pdZt,pt0=p0,t0≤t≤T,(38)yt,t0,y0,yt0=y0,pt,t0,y0,p0,pt0=p0,t0≤t<T,respectively.


## 5. Discrete-Time Dynamic Model for Local Sample Mean and Covariance Processes

In this section, we extend the discrete-time dynamic model for local sample mean and variance processes given in Otunuga et al. [16, 17] to a multivariate case including the jump process. The development of idea and model of statistic for mean and covariance processes was motivated by the state and parameter estimation problems of the continuous time nonlinear stochastic dynamic model of the energy commodity market network. For this purpose, we need to introduce a few definitions and notations. For more information, we direct the readers to the work of Otunuga et al. [16, 17].For eachi∈I1,K∗, let τi−1 and γ¯i−1 be finite constant time delays such that 0<γ¯i−1≤τi−1. If K=0, then there is no jump and we simply write τi−1=τ and γ¯i−1=γ¯. Here, τi−1 characterizes the influence of the past performance history of state of dynamic process. γ¯i−1 describes the reaction or response time delays. In general, these time delays are unknown and random variables. These types of delays play an important role in developing mathematical models of continuous time [30] and discrete-time [22, 31] dynamic processes. Based on the practical nature of data collection process, it is essential to either transform these time delays into positive integers or design a suitable data collection schedule or discretization process. For this purpose, for each i∈I1,K∗, we describe the discrete version of time delays of τi and γi as follows:
(39)ri−1=τi−1Δti−1+1,γ¯¯i−1=γ¯i−1Δti−1+1,fori∈I1,K∗.Here,. is the integer part of a real number, and for the sake of simplicity, we assume that 0<γ¯i−1<1,γ¯¯i−1=1.Definition 6.
Letx=x1,x2,⋯,xnΤ be a continuous time multivariate stochastic process defined on an interval t0−τ,T into Rn, for some T>0. For t∈t0−τ,T, let Ft be an increasing subsigma algebra of a complete probability space Ω,F,P for which xt is Ft measurable. For each i∈I1,K∗, let ℙ and ℙi−1 be a partition of t0−τ,T and Ti−1−τi−1,Ti, respectively. The partition ℙi−1 is derived by decomposing the partition ℙ. For each i∈I1,K∗, the partitions ℙ and ℙi−1 are defined as follows:
(40)ℙ=tk:tk=t0+kΔt,k∈I−r,N,ℙi−1=tki−1:tki−1=Ti−1+kΔt,k∈I−ri−1,Ni−1,where tki−1 represents time tk in the interval Ti−1−τi−1,Ti, Δt=T−t0/N≡Ti−Ti−1/Ni−1, and T0=t0.Remark 7.
We defineSi=∑l=1iNl−1 with S0=0. For K≠0, we note that we can write ℙ as t0<t1<⋯<tN0<tN0+1<⋯.<tN0+N1<tN0+N1+1<⋯.<tSi−1<tSi−1+1<⋯<tSi<⋯<T. From this, it follows directly that the jump times Ti are contained in ℙ. For any tki−1∈ℙi−1, k∈0,Ni−1, we have tki−1∈ℙ. Hence, there is a relationship between elements of ℙi−1 with ℙ that is described by tki−1=tSi−1+k, for k∈I0,Ni. In fact, the relationship between a set of jump times T1,T2,⋯,TK and the partition ℙ defined in (40) is as Ti→tSi, where Ni−1 is the size of partition ℙi−1 of the subinterval Ti−1,Ti. It follows that SK∗=N. Using these facts, and noting that if K=0, then ℙi−1=ℙ, Ni−1=N, τi−1=τ, γi−1=γ, ri−1=r, ηi−1=η, tki−1=tk and (40) reduces to
(41)ℙi−1=tki−1:tki−1=Ti−1+kΔt,k∈I−ri−1,Ni−1.
For eachi∈I1,K∗, let xi−1tki−1k=−ri−1Ni−1 be a finite sequence corresponding to the stochastic process x and partition ℙi−1 defined in (41). We simply write xtki−1≡xi−1tki−1. We further recall that xtki−1 is Ftki−1 measurable for k∈I−ri−1,Ni−1. We also recall the definition of forward time shift operator F [16, 17, 32]:
(42)Flxtki−1=xtk+li−1,l≥0.Definition 8.
Forqi−1=1 and ri−1≥1, each k∈I0,Ni−1 and each mki−1∈I2,ri−1+Si−1+k−1, a partition Pki−1 of closed interval tk−mki−1i−1,tk−1i−1 is called local at time tki−1, and it is defined by
(43)Pki−1≔tk−mki−1i−1<tk−mki−1+1i−1<⋯<tk−1i−1.
Moreover,Pki−1 is referred to as the mki−1 point subpartition of the partition ℙi−1 in (41) of the closed subinterval tk−mki−1i−1,tk−1i−1 of −τi−1,Ti.Remark 9.
We note that forK=0, that is, no jump, we have Pki−1=Pk, mki−1=mk, tk−mki−1i−1=tk−mki−1, and tk−1i−1=tk−1, where Pk is referred as the mk point subpartition of partition ℙ in (40) of the closed subinterval tk−mk,tk of t0−τ,T for k∈I0,N.Definition 10.
For eachi∈I1,K∗, k∈I0,Ni−1, and mki−1∈I2,ri−1+Si−1+k−1, a local finite sequence Smki−1,k at tki−1 of size mki−1 is the restriction [33] of xtki−1k=−ri−1Ni−1 to Pki−1 in (43). This restriction sequence is defined by
(44)Smki−1,k≔Flxtk−1i−1l=−mki−1+10.Asmki−1 varies from 2 to ri−1+Si−1+k−1, the corresponding respective local sequence Smki−1,k at tki−1 varies from xtli−1l=k−2k−1 to xtli−1l=−ri−1+Si−1+1k−1. As a result of this, the sequence defined in (44) is also called a mki−1 local moving sequence. Furthermore, the average corresponding to the local sequence Smki−1,k in (44) is defined by
(45)S¯mki−1,k=1mki−1∑l=−mki−1+10Flxtk−1i−1.The average/mean defined in (45) is also called the mki−1 local average/mean. For i∈I1,K∗ and k∈I0,Ni−1, the mki−1 local covariance matrix corresponding to the local sequence Smki−1,k in (44) is defined by
(46)∑mki−1,k=smki−1,k1,1smki−1,k1,2smki−1,k1,3⋯smki−1,k1,nsmki−1,k2,1smki−1,k2,2smki−1,k2,3⋯smki−1,k2,n⋮⋱⋱⋱⋮smki−1,kn,1smki−1,kn,2smki−1,kn,3⋯smki−1,kn,n,where smki−1,kj,l≡smki−1,kj,lx, j,l∈I1,n is the local sample covariance statistic between xj and xl at tki−1 described by
(47)smki−1,kj,l≔1mki−1∑a=−mki−1+10Faxjtk−1i−1−1mki−1∑b=−mki−1+10Fbxjtk−1i−1Faxltk−1i−1−1mki−1∑b=−mki−1+10Fbxltk−1i−1,forsmallmki−1,1mki−1−1∑a=−mki−1+10Faxjtk−1i−1−1mki−1∑b=−mki−1+10Fbxjtk−1i−1Faxltk−1i−1−1mki−1∑b=−mki−1+10Fbxltk−1i−1,forlargemki−1.Denotingxk≡xtki−1 for i∈I1,K∗ and k∈I1,Ni−1, the discrete-time interconnected dynamic model of local sample mean S¯mk,k and variance smk,k2 processes (DTIDMLSMVSP) [16, 17] is given by
(48)S¯mk−di−1+1i−1,k−di−1+1=mk−di−1i−1mk−di−1+1i−1S¯mk−di−1i−1,k−di−1+ηmk−di−1i−1,k−di−1,S¯mTi−1i−1,Ti−1=S¯Ti−1∑mki−1,k=mki−1−1mki−1∑j=1di−1mk−ji−1∏l=0j−1mk−li−1∑mk−ji−1,k−j+mk−di−1i−1∏l=0di−1−1mk−li−1S¯mk−di−1i−1,k−di−1S¯mk−di−1i−1,k−di−1Τ+εmk−1i−1,k−1,forsmallmki−1,mk−1i−1≤mki−1,∑j=1di−1mk−ji−1−1∏l=0j−1mk−li−1∑mk−ji−1,k−j+mk−di−1i−1∏l=0di−1−1mk−li−1S¯mk−di−1i−1,k−di−1S¯mk−di−1i−1,k−di−1Τ+εmk−1i−1,k−1forlargemki−1,mk−1i−1≤mki−1∑mji−1,j=∑j,i∈I1,K∗,j∈I−di−1,0,initialconditionswhere di−1 is the order of the system and
(49)η=η1η2⋮ηn,εmki−1,k=εmki−1,k1,1εmki−1,k1,2εmki−1,k1,3⋯εmki−1,k1,nεmki−1,k2,1εmki−1,k2,2εmki−1,k2,3⋯εmki−1,k2,n⋮⋱⋱⋱⋮εmki−1,kn,1εmki−1,kn,2εmki−1,kn,3⋯εmki−1,kn,n,ϵmki−1,k=εmki−1,k1,1εmki−1,k1,2εmki−1,k1,3⋯εmki−1,k1,nεmki−1,k2,1εmki−1,k2,2εmki−1,k2,3⋯εmki−1,k2,n⋮⋱⋱⋱⋮εmki−1,kn,1εmki−1,kn,2εmki−1,kn,3⋯εmki−1,kn,n,ηmk−di−1i−1,k−di−1j=1mk−di−1+1i−1∑ι=−mk−di−1+1i−1+1−mk−di−1i−1+1Fιxjk−di−1−F−mk−di−1i−1+1xjk−di−1−F−mk−di−1i−1xjk−di−1+F0xjk−di−1,εmk−1i−1,k−1j,l=mki−1−1mki−1∑ι=1di−1F−ι+1xjk−1F−ι+1xlk−1∏a=0ι−1mk−ai−1−∑ι=1di−1F−ι+1−mk−ιi−1xjk−1F−ι+1−mk−ιi−1xlk−1∏a=0ι−1mk−ai−1−∑ι=1di−1F−ι+2−mk−ιi−1xjk−1F−ι+2−mk−ιi−1xlk−1∏a=0ι−1mk−ai−1+mki−1−1mki−1∑ι=1di−1∑v=−ι+2−mk−ι+1i−1−ι+2−mk−ιi−1Fvxjk−1Fvxlk−1∏a=0ι−1mk−ai−1+∑ι=1di−1∑v,s=−ι+2−mk−ι+1i−1v≠s−ι+1Fvxjk−1Fsxlk−1∏a=0ι−1mk−ai−1−1mki−1∑v,s=−mki−1+1v≠s0Fvxjk−1Fsxlk−1,ϵmk−1i−1,k−1j,l=∑ι=1di−1F−ι+1xjk−1F−ι+1xlk−1∏a=0ι−1mk−ai−1−∑ι=1di−1F−ι+1−mk−ιi−1xjk−1F−ι+1−mk−ιi−1xlk−1∏a=0ι−1mk−ai−1−∑ι=1di−1F−ι+2−mk−ιi−1xjk−1F−ι+2−mk−ιi−1xlk−1∏a=0ι−1mk−ai−1+∑ι=1di−1∑v=−ι+2−mk−ι+1i−1−ι+2−mk−ιi−1Fvxjk−1Fvxlk−1∏a=0ι−1mk−ai−1+∑ι=1di−1∑v,s=−ι+2−mk−ι+1i−1v≠s−ι+1Fvxjk−1Fsxlk−1∏a=0ι−1mk−ai−1−1mki−1−1∑v,s=−mki−1+1v≠s0Fvxjk−1Fsxlk−1.The proof of this is given in [16].


## 6. Parametric Estimation

In this section, we consider a parameter estimation problem in drift and diffusion coefficients of (34). This is achieved by utilizing the lagged adaptive process [22] and the interconnected discrete-time dynamics of local sample mean and variances statistic processes model (48). For each i∈I1,K∗, we consider a general interconnected hybrid system described by the system of stochastic differential equations:
(50)dxi−1=fi−1t,xdt+σi−1t,xdWt,xTi−1=xi−1,t∈Ti−1,Ti,xi=Γixi−1Ti−,Ti−1,xi−1,where Γi=diagΓ1i,Γ2i,⋯,Γji,⋯,Γni is the jump coefficient matrix; the jump times Ti’s are defined in (34). For each j∈I1,n, the estimate of the jump coefficient Γji is given by Γji=xjTi/limt→Ti−xji−1t,Ti−1,xi−1.LetV∈C−τ,∞×Rn,Rm, and its partial derivatives Vt, ∂V/∂x, and ∂2V/∂x2 exist and are continuous on each interval Ti−1,Ti. We apply Itô-Doob stochastic differential formula [34] to V, and we obtain
(51)dVt,xi−1=LVt,xi−1dt+Vxt,xi−1σt,xi−1dWt,xTi−1=xi−1,t∈Ti−1,Ti,VTi,xi=VTi,Γixi−1Ti−,Ti−1,xi−1.where the L operator is defined by
(52)LVt,xi−1=Vtt,xi−1+Vxt,xi−1ft,xi−1+12trVxxt,xi−1ct,xi−1,ct,xi−1=σi−1t,xi−1σi−1t,xi−1⊤.For (50) and (51), we present the Euler-type discretization scheme [23]:
(53)Δxi−1tki−1=ftk−1i−1,xtk−1i−1Δtki−1+σi−1tk−1i−1,xtk−1i−1ΔWtki−1,k∈I1,Ni−1,xi=Γixi−1Ti−,Ti−1,xi−1,ΔVtki−1,xi−1tki−1=LVtk−1i−1,xi−1tk−1i−1Δtki−1+Vxtk−1i−1,xi−1tk−1i−1σi−1tk−1i−1,xtk−1i−1ΔWtki−1,VTi,xi=VTi,Γixi−1Ti−,Ti−1,xi−1.DefineFtk−1i−1≡Fk−1i−1 as the filtration process up to time tk−1i−1. With regard to the continuous time dynamic system (50) and its transformed system (51), the more general moments of Δxtki−1 are as follows:
(54)EΔxtki−1Fk−1i−1=fi−1tk−1i−1,xi−1tk−1i−1Δtki−1,EΔxi−1tki−1−EΔxi−1tki−1Fk−1i−1Δxi−1tki−1−EΔxi−1tki−1Fk−1i−1ΤFk−1i−1=σi−1tk−1i−1,xi−1tk−1i−1σi−1tk−1i−1,xi−1tk−1i−1ΤΔtki−1,EΔVtki−1,xi−1tki−1Fk−1i−1=LVtk−1i−1,xi−1tki−1,Δtki−1,EΔVtki−1,xi−1tki−1−EΔVtki−1,xi−1tki−1Fk−1i−1ΔVtki−1,xi−1tki−1−EΔVtki−1,xi−1tki−1Fk−1i−1ΤFk−1i−1=Btk−1i−1,xi−1tk−1i−1,where Btk−1i−1,xtk−1i−1=Vxtk−1i−1,xi−1tk−1i−1ctk−1i−1,xtk−1i−1Vxtk−1i−1,xtk−1i−1⊤Δtki−1, and Τ stands for the transpose of the matrix. From (53) and (54), we have
(55)Δxi−1tki−1=EΔxi−1tki−1Fk−1i−1+σi−1tk−1i−1,xi−1tk−1i−1ΔWtki−1,k∈I1,Ni−1,ΔVtki−1,xi−1tki−1=EΔVtki−1,xi−1tki−1Fk−1i−1+Vxi−1tk−1i−1,xi−1tk−1i−1σi−1tk−1i−1,xi−1tk−1i−1ΔWtk.This provides the basis for the development of the concept of lagged adaptive expectation process [22] with respect to continuous time stochastic dynamic system (51). This indeed leads to a formulation of mki−1 local generalized method of moments at tki−1.In the following, we state a result that exhibits the existence of the solution of the system of nonlinear algebraic equations. For the sake of easy reference, we shall state the implicit function theorem without proof.Theorem 11.
Implicit Function Theorem [33]. Let F=F1,F2,⋯,Fq be a vector-valued function defined on an open set S∈Rq+k with values in Rq. Suppose F∈C′ on S. Let (u0;v0) be a point in S for which Fu0;v0=0 and for which the q×q determinant of the Jacobian matrix det JFv0≠0. Then, there exists a k-dimensional open set T0 containing v0 and unique vector-valued function g, defined on T0 and having values in Rq, such that g∈C′ on T0, gv0=u0, and Fgv;v=0 for every v∈T0.


### 6.1. Special Case: Illustration

For eachj,l∈I1,n and each i∈I1,K∗, we consider a special case of (14).
(56)dyj=uji−1−yjκj,ji−1yj+∑l≠jnκj,li−1yldt+δj,ji−1uji−1−yjdWj,jt+uji−1−yj∑l≠jnδj,li−1yldWj,lt,yjTi−1=yji−1,t∈Ti−1,Ti,yji=πjiyji−1Ti−,Ti−1,yi−1,dpjt=pjγj,ji−1yj−pj+βji−1+∑l≠jnγj,li−1pltdt+σj,ji−1pjdZj,jt+pj∑l≠jnσj,li−1pldZj,lt,pjTi−1=pji−1,t∈Ti−1,Ti,pji=θjipji−1Ti−,Ti−1,yi−1,pi−1.Here,κj,li−1, uji−1, βji−1, γj,li−1, δj,li−1, and σj,li−1 are the system parameters on the jump subinterval Ti−1,Ti; uji−1, κj,ji−1, γj,ji−1, δj,ji−1, and σj,ji−1 are positive; and for l≠j, κj,li−1,γj,li−1∈R; δj,li−1, σj,li−1 are nonnegative. W and Z are independent standard Wiener process on a filtered probability space (Ω,F,Ftt≥0,P) with the properties described in (14). It follows that the interconnected system of stochastic differential equations (56) has 4n2+2n parameters. Also,
(57)κj,ll≠ji−1>0ifyliscooperatingwithyj,<0ifyliscompetingwithyj,=0ifthereisnointeractionbetweenylandyj,j,l∈I1,n,and likewise,
(58)γj,ll≠ji−1>0ifpliscooperatingwithpj,<0ifpliscompetingwithpj,=0ifthereisnointeractionbetweenplandpj,j,l∈I1,n.Remark 12.
For the caseK=0, (56) is reduced to
(59)dyj=uj−yjκj,jyj+∑l≠jnκj,lyldt+δj,juj−yjdWj,jt+uj−yj∑l≠jnδj,lyldWj,lt,yjt0=yj0,t∈t0,T,dpjt=pjγj,jyj−pj+βj+∑l≠jnγj,lpltdt+σj,jpjdZj,jt+pj∑l≠jnσj,lpldZj,lt,pjt0=pj0,t∈t0,T,where for j,l∈I1,n, the parameters κj,l, uj, βj, γj,l, δj,l, and σj,l are the system parameters on the interval t0,T; uj, κj,j, γj,j, δj,j, and σj,j are positive; and for l≠j, κj,l,γj,l∈R; δj,l, σj,l are nonnegative. For each j∈I1,n, we pick a Lyapunov function
(60)V1jt,yj=yjq,V2jt,pj=pjq,q∈ℤ+,in (51) for (56). Using the Itô differential formula [24], we have
(61)dV1j=qyjq−1uji−1−yjκj,ji−1yj+∑l≠jnκj,li−1yl+12qq−1yjq−2uji−1−yj2δj,ji−12+∑l≠jnδj,li−12yl2dt+qyjq−1uji−1−yjδj,ji−1dWj,jt+∑l≠jnδj,li−1yldWj,lt,yjTi−1=yji−1,t∈Ti−1,TiV1ji=πjiqyjTi−,Ti−1,yi−1q,ift=Ti,dV2j=pjqqγji−1yj−pj+βji−1+∑l≠jnγj,li−1pl+12qq−1σj,ji−12+∑l≠jnσj,li−12pl2dt+qpjqσj,ji−1dZj,jt+∑l≠jnσj,li−1pldZj,lt,pjTi−1=pji−1,t∈Ti−1,Ti,V2ji=θjiqpjTi−,Ti−1,yi−1,pi−1q,ift=Ti,
By settingΔtki−1=tki−1−tk−1i−1=Δt, Δytki−1=ytki−1−ytk−1i−1, and Δptki−1=ptki−1−ptk−1i−1, the combined Euler discretized scheme for (61) is
(62)Δyjqtki−1=qyjq−1tk−1i−1uji−1−yjtk−1i−1κj,ji−1yjtk−1i−1+∑l≠jnκj,li−1yltk−1i−1+12qq−1yjq−2tk−1i−1uji−1−yjtk−1i−12δj,ji−12+∑l≠jnδj,li−12yl2tk−1i−1Δt+qyjq−1tk−1i−1uji−1−yjtk−1i−1δj,ji−1ΔWj,jtki−1+∑l≠jnδj,li−1ylΔWj,ltki−1,yjTi−1=yji−1,tki−1∈Ti−1,Ti,yjiq=πjiqyjTi−,Ti−1,yi−1q,ift=Ti,Δpjqtki−1=pjqtk−1i−1qγj,ji−1yjtk−1i−1−pjtk−1i−1+βji−1+∑l≠jnγj,li−1pltk−1i−1+12qq−1σj,ji−12+∑l≠jnσj,li−12pl2tk−1i−1Δt+qpjqtk−1i−1σj,ji−1ΔZj,jtki−1+∑l≠jnσj,li−1pltk−1i−1ΔZj,ltki−1,pjTi−1=pji−1tki−1∈Ti−1,Ti,q∈I1,n+1,pjiq=θjiqpjTi−,Ti−1,yi−1,pi−1q,ift=Ti.where ytki−1k=−ri−10, ptki−1k=−ri−10 are given finite sequence of FTi−1i−1 measurable random vectors and are independent of ΔWtki−1k=0Ni−1, ΔZtki−1k=0Ni−1, respectively. We define Δyjqtki−1=yjqtki−1−yjqtk−1i−1 and Δpjqtki−1=pjqtki−1−pjqtk−1i−1.
Applying conditional expectation to (62) with respect to Ftk−1i−1≡Fk−1i−1, we obtain
(63)EΔyjqtki−1Fk−1i−1=qyjq−1tk−1i−1uji−1−yjtk−1i−1κj,ji−1yjtk−1i−1+∑l≠jnκj,li−1yltk−1i−1+qq−12Δtiyjq−2tk−1i−1EΔyjtki−1−EΔyjtki−1Fk−1i−12Fk−1i−1·Δtfortki−1∈Ti−1,Ti,Eyjiqtki−1Fk−1i−1=πjiqyjqTi−,Ti−1,yi−1,iftki−1=Ti,EΔpjqtki−1Fk−1i−1=qpjqtk−1i−1γj,ji−1yjtk−1i−1−pjtk−1i−1+βji−1+∑l≠jnγj,li−1pltk−1i−1+qq−12Δtipjq−2tk−1i−1EΔpjtki−1−EΔpjtki−1Fk−1i−12Fk−1i−1·Δt,fortki−1∈Ti−1,Ti,EpjiqFi−1=θjiqpjqTi−,Ti−1,yi−1,pi−1,iftki−1=Ti,q∈I1,n+1,(64)EΔyjqtki−1−EΔyjqtki−1Fk−1i−1Δylqtki−1−EΔylqtki−1Fk−1i−1Fk−1i−1=q2yjylq−1tk−1i−1uji−1−yjtk−1i−1uli−1−yltk−1i−1δj,ji−1δl,ji−1yjtk−1i−1+δl,li−1δj,li−1yltk−1i−1+∑r=1,j,l≠rnδj,ri−1δl,ri−1yr2tk−1i−1Δt,tki−1∈Ti−1−τi−1,Ti,EΔpjqtki−1−EΔpjqtki−1Fk−1i−1Δplqtki−1−EΔplqtki−1Fk−1i−1Fk−1i−1=q2pjplqtk−1i−12σj,ji−1σl,ji−1yjtk−1i−1+∑r=1,j,l≠rnσj,ri−1σl,ri−1yr2tk−1i−1,q,j≠l∈I1,n,where Fk−1i−1 is the filtration up to time tk−1i−1. From (62), (63) and (64) are reduced to
(65)Δyjqtki−1=EΔyjqtki−1Fk−1i−1+qyjq−1tk−1i−1uji−1−yjtk−1i−1δj,ji−1ΔWj,jtki−1+∑l≠jnδj,li−1ylΔWj,ltki−1,yjTi−1=yji−1,tki−1∈Ti−1,Ti,yjiq=πjiqyjqTi−,Ti−1,yi−1,iftki−1=Ti,Δpjqtki−1=EΔpjqtki−1Fk−1i−1+qpjqtk−1i−1σj,ji−1ΔZj,jtki−1+∑l≠jnσj,li−1pltk−1i−1ΔZj,ltki−1,pjTi−1=pji−1,tki−1∈Ti−1,Ti,q∈I1,n+1,j∈I1,n,pjiq=θjiqpjqTi−,Ti−1,yi−1,pi−1,iftki−1=Ti.Equation (65) provides the basis for the development of the concept of lagged adaptive expectation process [22] with respect to continuous time stochastic dynamic systems (56) and (61).Fork∈I0,Ni−1, applying the lagged adaptive expectation process [22] and Definitions 6, 8, and 10 and using (63), (64), and (65), we formulate a local observation/measurement process at tki−1 as an algebraic functions of mki−1 local functions of restriction of the finite sample sequence ytli−1l=−ri−1Ni−1 and ptli−1l=−ri−1Ni−1 to subpartition Pki−1 in Definition 8:
(66)1mki−1∑ι=k−mki−1k−1EΔyjqtιi−1Fι−1i−1=1mki−1∑ι=k−mki−1k−1qyjq−1tι−1i−1uji−1−yjtι−1i−1κj,ji−1yjtι−1i−1+∑l≠jnκj,li−1yltι−1i−1+qq−12Δtyjq−2tι−1i−1smki−1,k,Δyjj,jΔt,1mki−1∑ι=k−mki−1k−1EΔpjqtιi−1∣Fι−1i−1=1mki−1∑ι=k−mki−1k−1qpjqtι−1i−1γj,ji−1yjtι−1i−1−pjtι−1i−1+βji−1+∑l≠jnγj,lpltι−1i−1+qq−12Δtpjq−2tι−1i−1smki−1,k,Δpjj,jΔt,q∈I1,n+1,(67)s^mki−1,kj,lΔyq=1mki−1∑ι=k−mki−1k−1q2yjylq−1tι−1i−1uji−1−yjtι−1i−1uli−1−yltι−1i−1δj,ji−1δl,ji−1yjtι−1i−1+δl,li−1δj,li−1yltk−1i−1+∑r=1j≠l≠rnδj,ri−1δl,ri−1yr2tι−1i−1,s^mki−1,kj,lΔpq=1mki−1∑ι=k−mki−1k−1q2pjplqtι−1i−1σj,ji−1σl,ji−1pjtι−1i−1+σl,li−1σj,li−1pltι−1i−1+∑r=1j≠l≠rnσj,ri−1σl,ri−1pr2tι−1i−1,j≠l∈I1,n,q∈I1,2n.For eachi∈I1,K∗ and each j≠l∈I1,n, we define
(68)F1quji−1,κj,ri−1r=1n≡F1qEΔyjqtιi−1Fι−1i−1;uji−1,κj,ri−1r=1n,F2qδj,ri−1,δl,ri−1r=1n≡F2qEΔyjqtιi−1Fι−1i−1,EΔylqtιi−1Fι−1i−1;δj,ri−1,δl,ri−1r=1n,G1qβji−1,γj,ri−1r=1n≡G1qEΔpjqtιi−1∣Fι−1i−1;βji−1,γj,ri−1r=1n,G2qσj,ri−1,σl,ri−1r=1n≡G2qEΔpjqtιi−1∣Fι−1i−1,EΔplqtιi−1∣Fι−1i−1;σj,r,σl,ri−1r=1n,by
(69)F1quji−1,κj,ri−1r=1n=1mki−1∑ι=k−mki−1k−1qyjq−1tι−1i−1uji−1−yjtι−1i−1∑r=1nκj,ryrtι−1i−1+qq−12Δtyjq−2tι−1i−1s^mki−1,kj,jΔyjΔt−1mki−1∑ι=k−mki−1k−1EΔyjqtιi−1Fι−1i−1,q∈I1,n+1,F2qδj,ri−1,δl,ri−1r=1n=1mki−1∑ι=k−mki−1k−1q2yjylq−1tι−1i−1uji−1−yjtι−1i−1×uli−1−yltι−1i−1δj,ji−1δl,ji−1yjtι−1i−1+δl,li−1δj,li−1yltι−1i−1+∑r=1j≠l≠rnδj,ri−1δl,ri−1yr2tι−1i−1−s^mki−1,k,j,lΔyq,j≠l∈I1,n,q∈I1,2n,G1qβji−1,γj,ri−1r=1n=1mki−1∑ι=k−mki−1k−1qpjqtι−1i−1γj,ji−1yjtι−1i−1−pjtι−1i−1+βji−1+∑r≠jnγj,ri−1prtι−1i−1+qq−12Δtpjq−2tι−1i−1s^mki−1,kj,jΔpjΔt−1mki−1∑ι=k−mki−1k−1EΔpjqtιi−1∣Fι−1i−1,q∈I1,n+1,G2qσj,ri−1,σl,ri−1r=1n=1mki−1∑ι=k−mki−1k−1q2pjplqtι−1i−1σj,ji−1σl,ji−1pjtι−1i−1+σl,li−1σj,li−1pltι−1i−1+∑r=1j≠l≠rnσj,ri−1σl,ri−1pr2tι−1i−1−s^mki−1,k,Δpqj,l,j≠l∈I1,n,q∈I1,2n.For everyj∈I1,n, we have
(70)F1quji−1,κj,ri−1r=1n=0,q∈I1,n+1,F2qδj,ri−1,δl,ri−1r=1n=0,q∈I1,2n,G1qβji−1,γj,ri−1r=1n=0,q∈I1,n+1,G2qδj,ri−1,δl,ri−1r=1n=0,q∈I1,2n.We defineF1=F1qq∈I1,n+1, F2=F2qq∈I1,n, G1=G1qq∈I1,n+1, and G2=G2qq∈I1,n. Thus, provided that the determinant of each of the Jacobian matrices JF1uji−1,κj,ri−1r=1n, JF2uji−1,δj,ri−1r=1n, JG1βji−1,γj,ri−1r=1n and JG2σj,ri−1,σl,ri−1r=1n are not zero, by the application of Theorem 11 (Implicit Function Theorem), we conclude that for every nonconstant mki−1-local sequence yjtli−1l=k−mki−1k−1 and pjtli−1l=k−mki−1k−1, there exist a unique solution (u^ji−1mki−1,tki−1, β^ji−1mki−1,tki−1, κ^j,ri−1r=1nmki−1,tki−1, γ^j,ri−1r=1nmki−1,tki−1, δ^j,ri−1r=1nmki−1,tki−1, σ^j,ri−1r=1nmki−1,tki−1) of system of algebraic equations (70) as a point estimates of uji−1, κj,ri−1r=1n, γj,ri−1r=1n, δj,ri−1r=1n, σj,ri−1r=1n, j∈I1,n, respectively. We illustrate this approach using energy commodities natural gas, crude oil and coal [35–37] in the next section.


### 6.2. Illustration: Application to Energy Commodity

In this subsection, we present an illustration of the derived model on the natural gas, crude oil and coal [35–37]. Here, j∈I1,3 and i∈I1,K∗. Moreover, (56) reduces to
(71)dyj=uji−1−yjκj,ji−1yj+∑l≠j3κj,li−1yldt+δj,ji−1uji−1−yjdWj,jt+uji−1−yj∑l≠j3δj,li−1yldWj,lt,yjTi−1=yji−1,t∈Ti−1,Ti,yji=πjiyjTi−,Ti−1,yi−1,dpjt=pjγj,ji−1yj−pj+βji−1+∑l≠j3γj,li−1pltdt+σj,ji−1pjdZj,jt+pj∑l≠j3σj,li−1pldZj,lt,pjTi−1=pji−1,t∈Ti−1,Ti,pji=θjipjTi−,Ti−1,yi−1,pi−1.For eachj∈I1,3, following the argument used in Section 6.1, we have
(72)F1quji−1,κj,ri−1r=13=1mki−1∑ι=k−mki−1k−1qyjq−1tι−1i−1uji−1−yjtι−1i−1∑r=13κj,ryrtι−1i−1+qq−12Δtyjq−2tι−1i−1s^mki−1,kj,jΔyjΔt−1mki−1∑ι=k−mki−1k−1EΔyjqtιi−1Fι−1i−1,q∈I1,4,F2qδj,ri−1,δl,ri−1r=13=1mki−1∑ι=k−mki−1k−1q2yjylq−1tι−1i−1uji−1−yjtι−1i−1×uli−1−yltι−1i−1δj,ji−1δl,ji−1yjtι−1i−1+δl,li−1δj,li−1yltι−1i−1+∑r=1j≠l≠r3δj,ri−1δl,ri−1yr2tι−1i−1−s^mki−1,k,j,lΔyq,j≠l∈I1,3,q∈I1,6,(73)G1qβji−1,γj,ri−1r=13=1mki−1∑ι=k−mki−1k−1qpjqtι−1i−1γj,ji−1yjtι−1i−1−pjtι−1i−1+βji−1+∑r≠j3γj,ri−1prtι−1i−1+qq−12Δtpjq−2tι−1i−1s^mki−1,kj,jΔpjΔt−1mki−1∑ι=k−mki−1k−1EΔpjqtιi−1∣Fι−1i−1,q∈I1,4,G2qσj,ri−1,σl,ri−1r=13=1mki−1∑ι=k−mki−1k−1q2pjplqtι−1i−1σj,ji−1σl,ji−1pjtι−1i−1+σl,li−1σj,li−1pltι−1i−1+∑r=1j≠l≠r3σj,ri−1σl,ri−1pr2tι−1i−1−s^mki−1,k,Δpqj,l,j≠l∈I1,3,q∈I1,6,F1quji−1,κj,ri−1r=13=0,q∈I1,4,F2qδj,ri−1,δl,ri−1r=13=0,q∈I1,6,G1qβji−1,γj,ri−1r=13=0,q∈I1,4,G2qδj,ri−1,δl,ri−1r=13=0,q∈I1,6.We also haveF1=F1qq∈I1,4, F2=F2qq∈I1,3, G1=G1qq∈I1,4, and G2=G2qq∈I1,3. For each j∈I1,3, it follows that the determinant of the Jacobian of F1uji−1,κj,ri−1r∈I1,3, that is,JF1uji−1,κj,ri−1r∈I1,3, is not zero provided that all parameters κj,rr∈I1,3 are not zero or provided the sequence yji−1tri−1r=k−mki−1−1k−1 is neither zero nor constant. Likewise, determinants of the Jacobians JF2uji−1,δj,ri−1r∈I1,n, JG1βji−1,γj,ri−1r∈I1,n and JG2σj,ri−1r∈I1,n are nonzero if δj,rj,l∈I1,3, γj,rj,l∈I1,3, and σj,rj,l∈I1,3 are not zero or provided the sequence yji−1tri−1r=k−mki−1−1k−1 and pji−1tri−1r=k−mki−1−1k−1 are neither zero nor constant for j∈I1,3.Remark 13.
If the sampleyji−1tri−1r=k−mki−1−1k−1 is a constant sequence, it follows from (62) (q = 1) and the fact that Δyji−1tki−1=0 and smki−1,kj,jΔyj=0 that we can set uji−1mki−1,tki−1=1/mki−1∑r=k−mki−1k−1yji−1mki−1,tki−1. It also follows from (66) that κj,ri−1mki−1,tki−1=0.


## 6.1. Special Case: Illustration

For eachj,l∈I1,n and each i∈I1,K∗, we consider a special case of (14).
(56)dyj=uji−1−yjκj,ji−1yj+∑l≠jnκj,li−1yldt+δj,ji−1uji−1−yjdWj,jt+uji−1−yj∑l≠jnδj,li−1yldWj,lt,yjTi−1=yji−1,t∈Ti−1,Ti,yji=πjiyji−1Ti−,Ti−1,yi−1,dpjt=pjγj,ji−1yj−pj+βji−1+∑l≠jnγj,li−1pltdt+σj,ji−1pjdZj,jt+pj∑l≠jnσj,li−1pldZj,lt,pjTi−1=pji−1,t∈Ti−1,Ti,pji=θjipji−1Ti−,Ti−1,yi−1,pi−1.Here,κj,li−1, uji−1, βji−1, γj,li−1, δj,li−1, and σj,li−1 are the system parameters on the jump subinterval Ti−1,Ti; uji−1, κj,ji−1, γj,ji−1, δj,ji−1, and σj,ji−1 are positive; and for l≠j, κj,li−1,γj,li−1∈R; δj,li−1, σj,li−1 are nonnegative. W and Z are independent standard Wiener process on a filtered probability space (Ω,F,Ftt≥0,P) with the properties described in (14). It follows that the interconnected system of stochastic differential equations (56) has 4n2+2n parameters. Also,
(57)κj,ll≠ji−1>0ifyliscooperatingwithyj,<0ifyliscompetingwithyj,=0ifthereisnointeractionbetweenylandyj,j,l∈I1,n,and likewise,
(58)γj,ll≠ji−1>0ifpliscooperatingwithpj,<0ifpliscompetingwithpj,=0ifthereisnointeractionbetweenplandpj,j,l∈I1,n.Remark 12.
For the caseK=0, (56) is reduced to
(59)dyj=uj−yjκj,jyj+∑l≠jnκj,lyldt+δj,juj−yjdWj,jt+uj−yj∑l≠jnδj,lyldWj,lt,yjt0=yj0,t∈t0,T,dpjt=pjγj,jyj−pj+βj+∑l≠jnγj,lpltdt+σj,jpjdZj,jt+pj∑l≠jnσj,lpldZj,lt,pjt0=pj0,t∈t0,T,where for j,l∈I1,n, the parameters κj,l, uj, βj, γj,l, δj,l, and σj,l are the system parameters on the interval t0,T; uj, κj,j, γj,j, δj,j, and σj,j are positive; and for l≠j, κj,l,γj,l∈R; δj,l, σj,l are nonnegative. For each j∈I1,n, we pick a Lyapunov function
(60)V1jt,yj=yjq,V2jt,pj=pjq,q∈ℤ+,in (51) for (56). Using the Itô differential formula [24], we have
(61)dV1j=qyjq−1uji−1−yjκj,ji−1yj+∑l≠jnκj,li−1yl+12qq−1yjq−2uji−1−yj2δj,ji−12+∑l≠jnδj,li−12yl2dt+qyjq−1uji−1−yjδj,ji−1dWj,jt+∑l≠jnδj,li−1yldWj,lt,yjTi−1=yji−1,t∈Ti−1,TiV1ji=πjiqyjTi−,Ti−1,yi−1q,ift=Ti,dV2j=pjqqγji−1yj−pj+βji−1+∑l≠jnγj,li−1pl+12qq−1σj,ji−12+∑l≠jnσj,li−12pl2dt+qpjqσj,ji−1dZj,jt+∑l≠jnσj,li−1pldZj,lt,pjTi−1=pji−1,t∈Ti−1,Ti,V2ji=θjiqpjTi−,Ti−1,yi−1,pi−1q,ift=Ti,
By settingΔtki−1=tki−1−tk−1i−1=Δt, Δytki−1=ytki−1−ytk−1i−1, and Δptki−1=ptki−1−ptk−1i−1, the combined Euler discretized scheme for (61) is
(62)Δyjqtki−1=qyjq−1tk−1i−1uji−1−yjtk−1i−1κj,ji−1yjtk−1i−1+∑l≠jnκj,li−1yltk−1i−1+12qq−1yjq−2tk−1i−1uji−1−yjtk−1i−12δj,ji−12+∑l≠jnδj,li−12yl2tk−1i−1Δt+qyjq−1tk−1i−1uji−1−yjtk−1i−1δj,ji−1ΔWj,jtki−1+∑l≠jnδj,li−1ylΔWj,ltki−1,yjTi−1=yji−1,tki−1∈Ti−1,Ti,yjiq=πjiqyjTi−,Ti−1,yi−1q,ift=Ti,Δpjqtki−1=pjqtk−1i−1qγj,ji−1yjtk−1i−1−pjtk−1i−1+βji−1+∑l≠jnγj,li−1pltk−1i−1+12qq−1σj,ji−12+∑l≠jnσj,li−12pl2tk−1i−1Δt+qpjqtk−1i−1σj,ji−1ΔZj,jtki−1+∑l≠jnσj,li−1pltk−1i−1ΔZj,ltki−1,pjTi−1=pji−1tki−1∈Ti−1,Ti,q∈I1,n+1,pjiq=θjiqpjTi−,Ti−1,yi−1,pi−1q,ift=Ti.where ytki−1k=−ri−10, ptki−1k=−ri−10 are given finite sequence of FTi−1i−1 measurable random vectors and are independent of ΔWtki−1k=0Ni−1, ΔZtki−1k=0Ni−1, respectively. We define Δyjqtki−1=yjqtki−1−yjqtk−1i−1 and Δpjqtki−1=pjqtki−1−pjqtk−1i−1.
Applying conditional expectation to (62) with respect to Ftk−1i−1≡Fk−1i−1, we obtain
(63)EΔyjqtki−1Fk−1i−1=qyjq−1tk−1i−1uji−1−yjtk−1i−1κj,ji−1yjtk−1i−1+∑l≠jnκj,li−1yltk−1i−1+qq−12Δtiyjq−2tk−1i−1EΔyjtki−1−EΔyjtki−1Fk−1i−12Fk−1i−1·Δtfortki−1∈Ti−1,Ti,Eyjiqtki−1Fk−1i−1=πjiqyjqTi−,Ti−1,yi−1,iftki−1=Ti,EΔpjqtki−1Fk−1i−1=qpjqtk−1i−1γj,ji−1yjtk−1i−1−pjtk−1i−1+βji−1+∑l≠jnγj,li−1pltk−1i−1+qq−12Δtipjq−2tk−1i−1EΔpjtki−1−EΔpjtki−1Fk−1i−12Fk−1i−1·Δt,fortki−1∈Ti−1,Ti,EpjiqFi−1=θjiqpjqTi−,Ti−1,yi−1,pi−1,iftki−1=Ti,q∈I1,n+1,(64)EΔyjqtki−1−EΔyjqtki−1Fk−1i−1Δylqtki−1−EΔylqtki−1Fk−1i−1Fk−1i−1=q2yjylq−1tk−1i−1uji−1−yjtk−1i−1uli−1−yltk−1i−1δj,ji−1δl,ji−1yjtk−1i−1+δl,li−1δj,li−1yltk−1i−1+∑r=1,j,l≠rnδj,ri−1δl,ri−1yr2tk−1i−1Δt,tki−1∈Ti−1−τi−1,Ti,EΔpjqtki−1−EΔpjqtki−1Fk−1i−1Δplqtki−1−EΔplqtki−1Fk−1i−1Fk−1i−1=q2pjplqtk−1i−12σj,ji−1σl,ji−1yjtk−1i−1+∑r=1,j,l≠rnσj,ri−1σl,ri−1yr2tk−1i−1,q,j≠l∈I1,n,where Fk−1i−1 is the filtration up to time tk−1i−1. From (62), (63) and (64) are reduced to
(65)Δyjqtki−1=EΔyjqtki−1Fk−1i−1+qyjq−1tk−1i−1uji−1−yjtk−1i−1δj,ji−1ΔWj,jtki−1+∑l≠jnδj,li−1ylΔWj,ltki−1,yjTi−1=yji−1,tki−1∈Ti−1,Ti,yjiq=πjiqyjqTi−,Ti−1,yi−1,iftki−1=Ti,Δpjqtki−1=EΔpjqtki−1Fk−1i−1+qpjqtk−1i−1σj,ji−1ΔZj,jtki−1+∑l≠jnσj,li−1pltk−1i−1ΔZj,ltki−1,pjTi−1=pji−1,tki−1∈Ti−1,Ti,q∈I1,n+1,j∈I1,n,pjiq=θjiqpjqTi−,Ti−1,yi−1,pi−1,iftki−1=Ti.Equation (65) provides the basis for the development of the concept of lagged adaptive expectation process [22] with respect to continuous time stochastic dynamic systems (56) and (61).Fork∈I0,Ni−1, applying the lagged adaptive expectation process [22] and Definitions 6, 8, and 10 and using (63), (64), and (65), we formulate a local observation/measurement process at tki−1 as an algebraic functions of mki−1 local functions of restriction of the finite sample sequence ytli−1l=−ri−1Ni−1 and ptli−1l=−ri−1Ni−1 to subpartition Pki−1 in Definition 8:
(66)1mki−1∑ι=k−mki−1k−1EΔyjqtιi−1Fι−1i−1=1mki−1∑ι=k−mki−1k−1qyjq−1tι−1i−1uji−1−yjtι−1i−1κj,ji−1yjtι−1i−1+∑l≠jnκj,li−1yltι−1i−1+qq−12Δtyjq−2tι−1i−1smki−1,k,Δyjj,jΔt,1mki−1∑ι=k−mki−1k−1EΔpjqtιi−1∣Fι−1i−1=1mki−1∑ι=k−mki−1k−1qpjqtι−1i−1γj,ji−1yjtι−1i−1−pjtι−1i−1+βji−1+∑l≠jnγj,lpltι−1i−1+qq−12Δtpjq−2tι−1i−1smki−1,k,Δpjj,jΔt,q∈I1,n+1,(67)s^mki−1,kj,lΔyq=1mki−1∑ι=k−mki−1k−1q2yjylq−1tι−1i−1uji−1−yjtι−1i−1uli−1−yltι−1i−1δj,ji−1δl,ji−1yjtι−1i−1+δl,li−1δj,li−1yltk−1i−1+∑r=1j≠l≠rnδj,ri−1δl,ri−1yr2tι−1i−1,s^mki−1,kj,lΔpq=1mki−1∑ι=k−mki−1k−1q2pjplqtι−1i−1σj,ji−1σl,ji−1pjtι−1i−1+σl,li−1σj,li−1pltι−1i−1+∑r=1j≠l≠rnσj,ri−1σl,ri−1pr2tι−1i−1,j≠l∈I1,n,q∈I1,2n.For eachi∈I1,K∗ and each j≠l∈I1,n, we define
(68)F1quji−1,κj,ri−1r=1n≡F1qEΔyjqtιi−1Fι−1i−1;uji−1,κj,ri−1r=1n,F2qδj,ri−1,δl,ri−1r=1n≡F2qEΔyjqtιi−1Fι−1i−1,EΔylqtιi−1Fι−1i−1;δj,ri−1,δl,ri−1r=1n,G1qβji−1,γj,ri−1r=1n≡G1qEΔpjqtιi−1∣Fι−1i−1;βji−1,γj,ri−1r=1n,G2qσj,ri−1,σl,ri−1r=1n≡G2qEΔpjqtιi−1∣Fι−1i−1,EΔplqtιi−1∣Fι−1i−1;σj,r,σl,ri−1r=1n,by
(69)F1quji−1,κj,ri−1r=1n=1mki−1∑ι=k−mki−1k−1qyjq−1tι−1i−1uji−1−yjtι−1i−1∑r=1nκj,ryrtι−1i−1+qq−12Δtyjq−2tι−1i−1s^mki−1,kj,jΔyjΔt−1mki−1∑ι=k−mki−1k−1EΔyjqtιi−1Fι−1i−1,q∈I1,n+1,F2qδj,ri−1,δl,ri−1r=1n=1mki−1∑ι=k−mki−1k−1q2yjylq−1tι−1i−1uji−1−yjtι−1i−1×uli−1−yltι−1i−1δj,ji−1δl,ji−1yjtι−1i−1+δl,li−1δj,li−1yltι−1i−1+∑r=1j≠l≠rnδj,ri−1δl,ri−1yr2tι−1i−1−s^mki−1,k,j,lΔyq,j≠l∈I1,n,q∈I1,2n,G1qβji−1,γj,ri−1r=1n=1mki−1∑ι=k−mki−1k−1qpjqtι−1i−1γj,ji−1yjtι−1i−1−pjtι−1i−1+βji−1+∑r≠jnγj,ri−1prtι−1i−1+qq−12Δtpjq−2tι−1i−1s^mki−1,kj,jΔpjΔt−1mki−1∑ι=k−mki−1k−1EΔpjqtιi−1∣Fι−1i−1,q∈I1,n+1,G2qσj,ri−1,σl,ri−1r=1n=1mki−1∑ι=k−mki−1k−1q2pjplqtι−1i−1σj,ji−1σl,ji−1pjtι−1i−1+σl,li−1σj,li−1pltι−1i−1+∑r=1j≠l≠rnσj,ri−1σl,ri−1pr2tι−1i−1−s^mki−1,k,Δpqj,l,j≠l∈I1,n,q∈I1,2n.For everyj∈I1,n, we have
(70)F1quji−1,κj,ri−1r=1n=0,q∈I1,n+1,F2qδj,ri−1,δl,ri−1r=1n=0,q∈I1,2n,G1qβji−1,γj,ri−1r=1n=0,q∈I1,n+1,G2qδj,ri−1,δl,ri−1r=1n=0,q∈I1,2n.We defineF1=F1qq∈I1,n+1, F2=F2qq∈I1,n, G1=G1qq∈I1,n+1, and G2=G2qq∈I1,n. Thus, provided that the determinant of each of the Jacobian matrices JF1uji−1,κj,ri−1r=1n, JF2uji−1,δj,ri−1r=1n, JG1βji−1,γj,ri−1r=1n and JG2σj,ri−1,σl,ri−1r=1n are not zero, by the application of Theorem 11 (Implicit Function Theorem), we conclude that for every nonconstant mki−1-local sequence yjtli−1l=k−mki−1k−1 and pjtli−1l=k−mki−1k−1, there exist a unique solution (u^ji−1mki−1,tki−1, β^ji−1mki−1,tki−1, κ^j,ri−1r=1nmki−1,tki−1, γ^j,ri−1r=1nmki−1,tki−1, δ^j,ri−1r=1nmki−1,tki−1, σ^j,ri−1r=1nmki−1,tki−1) of system of algebraic equations (70) as a point estimates of uji−1, κj,ri−1r=1n, γj,ri−1r=1n, δj,ri−1r=1n, σj,ri−1r=1n, j∈I1,n, respectively. We illustrate this approach using energy commodities natural gas, crude oil and coal [35–37] in the next section.


## 6.2. Illustration: Application to Energy Commodity

In this subsection, we present an illustration of the derived model on the natural gas, crude oil and coal [35–37]. Here, j∈I1,3 and i∈I1,K∗. Moreover, (56) reduces to
(71)dyj=uji−1−yjκj,ji−1yj+∑l≠j3κj,li−1yldt+δj,ji−1uji−1−yjdWj,jt+uji−1−yj∑l≠j3δj,li−1yldWj,lt,yjTi−1=yji−1,t∈Ti−1,Ti,yji=πjiyjTi−,Ti−1,yi−1,dpjt=pjγj,ji−1yj−pj+βji−1+∑l≠j3γj,li−1pltdt+σj,ji−1pjdZj,jt+pj∑l≠j3σj,li−1pldZj,lt,pjTi−1=pji−1,t∈Ti−1,Ti,pji=θjipjTi−,Ti−1,yi−1,pi−1.For eachj∈I1,3, following the argument used in Section 6.1, we have
(72)F1quji−1,κj,ri−1r=13=1mki−1∑ι=k−mki−1k−1qyjq−1tι−1i−1uji−1−yjtι−1i−1∑r=13κj,ryrtι−1i−1+qq−12Δtyjq−2tι−1i−1s^mki−1,kj,jΔyjΔt−1mki−1∑ι=k−mki−1k−1EΔyjqtιi−1Fι−1i−1,q∈I1,4,F2qδj,ri−1,δl,ri−1r=13=1mki−1∑ι=k−mki−1k−1q2yjylq−1tι−1i−1uji−1−yjtι−1i−1×uli−1−yltι−1i−1δj,ji−1δl,ji−1yjtι−1i−1+δl,li−1δj,li−1yltι−1i−1+∑r=1j≠l≠r3δj,ri−1δl,ri−1yr2tι−1i−1−s^mki−1,k,j,lΔyq,j≠l∈I1,3,q∈I1,6,(73)G1qβji−1,γj,ri−1r=13=1mki−1∑ι=k−mki−1k−1qpjqtι−1i−1γj,ji−1yjtι−1i−1−pjtι−1i−1+βji−1+∑r≠j3γj,ri−1prtι−1i−1+qq−12Δtpjq−2tι−1i−1s^mki−1,kj,jΔpjΔt−1mki−1∑ι=k−mki−1k−1EΔpjqtιi−1∣Fι−1i−1,q∈I1,4,G2qσj,ri−1,σl,ri−1r=13=1mki−1∑ι=k−mki−1k−1q2pjplqtι−1i−1σj,ji−1σl,ji−1pjtι−1i−1+σl,li−1σj,li−1pltι−1i−1+∑r=1j≠l≠r3σj,ri−1σl,ri−1pr2tι−1i−1−s^mki−1,k,Δpqj,l,j≠l∈I1,3,q∈I1,6,F1quji−1,κj,ri−1r=13=0,q∈I1,4,F2qδj,ri−1,δl,ri−1r=13=0,q∈I1,6,G1qβji−1,γj,ri−1r=13=0,q∈I1,4,G2qδj,ri−1,δl,ri−1r=13=0,q∈I1,6.We also haveF1=F1qq∈I1,4, F2=F2qq∈I1,3, G1=G1qq∈I1,4, and G2=G2qq∈I1,3. For each j∈I1,3, it follows that the determinant of the Jacobian of F1uji−1,κj,ri−1r∈I1,3, that is,JF1uji−1,κj,ri−1r∈I1,3, is not zero provided that all parameters κj,rr∈I1,3 are not zero or provided the sequence yji−1tri−1r=k−mki−1−1k−1 is neither zero nor constant. Likewise, determinants of the Jacobians JF2uji−1,δj,ri−1r∈I1,n, JG1βji−1,γj,ri−1r∈I1,n and JG2σj,ri−1r∈I1,n are nonzero if δj,rj,l∈I1,3, γj,rj,l∈I1,3, and σj,rj,l∈I1,3 are not zero or provided the sequence yji−1tri−1r=k−mki−1−1k−1 and pji−1tri−1r=k−mki−1−1k−1 are neither zero nor constant for j∈I1,3.Remark 13.
If the sampleyji−1tri−1r=k−mki−1−1k−1 is a constant sequence, it follows from (62) (q = 1) and the fact that Δyji−1tki−1=0 and smki−1,kj,jΔyj=0 that we can set uji−1mki−1,tki−1=1/mki−1∑r=k−mki−1k−1yji−1mki−1,tki−1. It also follows from (66) that κj,ri−1mki−1,tki−1=0.


## 7. Computational Algorithm

In this section, we outline computational, data organizational, and simulation schemes. We introduce the ideas of iterative data process and data simulation time schedules in relation to the real-time data observation/collection schedule. For the computational estimation of continuous time stochastic dynamic system state and parameters, it is essential to identify an admissible set of local conditional sample average and sample covariance parameters, namely, the size of local conditional sample in the context of a partition of time intervalTi−1−τi−1,Ti. Moreover, the discrete-time dynamic model of conditional sample mean and sample covariance statistic processes in Section 5 and the theoretical parameter estimation scheme in Section 6 motivates to outline a computational scheme in a systematic and coherent manner. A brief conceptual computational scheme and simulation process summary is described below.


### 7.1. Coordination of Data Observation, Iterative Process, and Simulation Schedules

Without loss of generality, we assume that the real data observation/collection partition schedulesℙi−1, i∈I1,K∗ are defined in (41). We present definitions of iterative process and simulation time schedule following the definitions in Otunuga et al. [16].Definition 14.
The iterative process time schedule in relation with the real data collection schedule is defined by(74)Iℙi−1=F−ri−1tki−1:fortki−1∈ℙi−1,fori∈I1,K∗,k∈I−ri−1,Ni−1,where F−ri−1tki−1=tk−ri−1i−1 is a forward shift operator [32].
The simulation time is based on the orderdi−1 of the time series model of mki−1 local conditional sample mean and covariance processes in (48).Remark 15.
For the case whereK=0, we have Iℙi−1=Iℙ, where ℙi−1=ℙ is defined in (40). This is the iterative time schedule in the absence of jumps.Definition 16.
The simulation process time schedule in relation with the real data observation schedule is defined by(75)Sℙi−1=Fri−1tki−1:fortki−1∈ℙi−1,ifdi−1≤ri−1,Fdi−1tki−1:fortki−1∈ℙi−1,ifdi−1>ri−1,k∈I−ri−1,Ni−1.Remark 17.
For eachi∈I1,K∗, the initial times of iterative and simulation processes are equal to the real data times tri−1i−1 and tdi−1i−1, whenever di−1≤ri−1 and di−1>ri−1, respectively. The iterative process and simulation process times with jump are tk+ri−1i−1 and tk+di−1i−1, i∈I1,K∗, respectively.


### 7.2. Conceptual Computational Parameter Estimation Scheme

For the conceptual computational dynamic system parameter estimation, we need to introduce a few concepts of local admissible sample/data observation sizemki−1 local admissible conditional finite sequence at tki−1∈Sℙi−1 and local finite sequence of parameter estimates at tki−1.Definition 18.
For eachi∈I1,K∗, and tki−1∈ITi−1−τi−1,Ti, we define local admissible sample/data observation size mki−1 at tki−1 as mki−1∈OSki−1, where
(76)OSki−1=I2,ri−1+Si−1+k−1,ifdi−1≤ri−1,I2,di−1+Si−1+k−1,ifdi−1>ri−1,k∈I0,Ni−1.
Moreover,OSki−1 is referred as the local admissible set of lagged sample/data observation size at tki−1.Remark 19.
We note that ifK=0, Si−1=0, the point tki−1=tk∈t0,T. Thus, (76) is reduced to
(77)OSki−1=OSk=I2,r+k−1,ifd≤r,I2,d+k−1,ifd>r,k∈I0,N.Definition 20.
For eachi∈I1,K∗, mki−1∈OSki−1 in Definition 18 and k∈I0,Ni−1, a mki−1 local admissible lagged adapted finite restriction sequence of conditional sample/data observation at time tki−1 to subpartition Pki−1 of ℙi−1 in Definition 8 is defined by Eyi−1tli−1Fl−1i−1l=k−mki−1k−1,Epi−1tli−1Fl−1i−1l=k−mki−1k−1. Moreover, a mki−1 class of admissible lagged adapted finite sequences of conditional sample/data observation of size mki−1 at tki−1 is defined by
(78)ASki−1=Eyi−1tli−1Fl−1i−1l=k−mki−1k−1mki−1∈OSki−1,Epi−1tli−1Fl−1i−1l=k−mki−1k−1mki−1∈OSki−1.In the case of energy commodity model, for eachi∈I1,K∗, mki−1∈OSki−1, we find corresponding mki−1 local admissible adapted finite sequence of conditional sample/data observation at tki−1, Eyi−1tli−1Fl−1i−1l=k−mki−1k−1,Epi−1tli−1Fl−1i−1l=k−mki−1k−1. For i∈I1,K∗, using this sequence and solutions of (70), we compute
(79)uji−1mk,tki−1,βji−1mki−1,tki−1,κj,li−1mki−1,tki−1,γj,li−1mki−1,tki−1,δj,li−1mki−1,tki−1,σj,li−1mki−1,tki−1,k∈0,Ni−1,for j,l∈I1,n. This leads to a local finite sequence of parameter estimates at tki−1 defined on OSki−1 as follows:
(80)u^ji−1mki−1,tki−1,β^ji−1mki−1,tki−1,κ^j,li−1mki−1,tki−1,γ^j,li−1mki−1,tki−1,δ^j,li−1mki−1,tki−1,σ^j,li−1mki−1,tki−1mki−1∈OSki−1.The above defined collection is denoted by(81)Uk,Bk,Kk,γk,δk,σk=u^ji−1mki−1,tki−1,β^ji−1mki−1,tki−1,κ^j,li−1mki−1,tki−1,γ^j,li−1mki−1,tki−1,δ^j,li−1mki−1,tki−1,σ^j,li−1mki−1,tki−1mki−1∈OSki−1,forj∈I1,n,i∈I1,K∗.


### 7.3. Conceptual Computation of State Simulation Scheme

For the development of a conceptual computational scheme, we need to employ the method of induction. The presented simulation scheme is based on the idea of lagged adaptive expectation process [17]. For j,l∈I1,n, an autocorrelation function (ACF) analysis [21, 32] performed on smki−1,kj,jy,smki−1,kj,jp suggests that the interconnected discrete-time dynamic model of local conditional sample mean and sample variance statistics in (48) is of order di−1=2. In view of this, we need to identify the initial data. We begin with a given initial data yi−1Ti−1,pi−1Ti−1, ∑m0i−1,t0i−1ym0i−1∈OS0i−1,∑m0i−1,t0i−1pm0i−1∈OS0i−1,∑m−1i−1,t−1i−1ym−1i−1∈OS−1i−1,∑m−1i−1,t−1i−1pm−1i−1∈OS−1i−1, S¯m−1i−1,t−1i−1ym−1i−1∈OS−1i−1,S¯m−1i−1,t−1i−1pm−1i−1∈OS−1i−1 as defined in (45), (46), (47), and (48).Letysmki−1,tki−1,psmki−1,tki−1≡yi−1,smki−1,tki−1,pi−1,smki−1,tki−1 be a simulated value of Eyi−1tki−1Fk−1i−1,Epi−1tki−1Fk−1i−1 at time tki−1 corresponding to an admissible sequence Eyi−1tli−1Fl−1i−1,Epi−1tli−1Fl−1i−1l=k−mkk−1∈ASki−1. For q=1, and j∈I1,n, the simulated value yjsmki−1,tki−1,pjsmki−1,tki−1 is generated from the discretized Euler scheme (62) as follows:
(82)yjsmki−1,tki−1=yjsmk−1i−1,tk−1i−1+uji−1mk−1i−1,tk−1i−1−yjsmk−1i−1,tk−1i−1κj,ji−1mk−1i−1,tk−1i−1yjsmk−1i−1,tk−1i−1+∑l≠jnκj,li−1mk−1i−1,tk−1i−1ylsmk−1i−1,tk−1i−1Δt+uji−1mk−1i−1,tk−1i−1−yjsmk−1i−1,tk−1i−1δj,ji−1mk−1i−1,tk−1i−1ΔWj,jmki−1,tki−1+∑l≠jnδj,li−1mk−1i−1,tk−1i−1ylsmk−1i−1,tk−1i−1ΔWj,lmki−1,tki−1,tki−1∈Ti−1,Ti,pjsmki−1,tki−1=pjsγj,ji−1mk−1i−1,tk−1i−1yjsmk−1i−1,tk−1i−1−pjsmk−1i−1,tk−1i−1+βji−1mk−1i−1,tk−1i−1+∑l≠jnγj,li−1mk−1i−1,tk−1i−1plsmk−1i−1,tk−1i−1Δt+σj,ji−1mk−1i−1,tk−1i−1pjsmk−1i−1,tk−1i−1ΔZj,jmki−1,tki−1+pjsmk−1i−1,tk−1i−1∑l≠jnσj,li−1mk−1i−1,tk−1i−1plsmk−1i−1,tk−1i−1ΔZj,lmki−1,tki−1,tki−1∈Ti−1,Ti.To find the simulated valueyji,sTi and pji,sTi, we need to estimate π^ji and θ^ji by first simulatinglimt→Ti−yji−1t,Ti−1,yi−1,s≡yjsmNi−1i−1,tNi−1i−1 and limt→Ti−pji−1t,Ti−1,yi−1,s,pi−1,s≡pjsmNi−1i−1,tNi−1i−1 as follows:
(83)yjsmNi−1i−1,tNi−1i−1=yjsmNi−1−1i−1,tNi−1−1i−1+uji−1mNi−1−1i−1,tNi−1−1i−1−yjsmNi−1−1i−1,tNi−1−1i−1∑l=1nκj,li−1mNi−1−1i−1,tNi−1−1i−1ylsmNi−1−1i−1,tNi−1−1i−1Δt+uji−1mNi−1−1i−1,tNi−1−1i−1−yjsmNi−1−1i−1,tNi−1−1i−1δj,ji−1mNi−1−1i−1,tNi−1−1i−1ΔWj,jmNi−1i−1,tNi−1i−1+∑l≠jnδj,li−1mNi−1−1i−1,tNi−1−1i−1ylsmNi−1−1i−1,tNi−1−1i−1ΔWj,lmNi−1i−1,tNi−1i−1,pjsmNi−1i−1,tNi−1i−1=pjsγj,ji−1mNi−1−1i−1,tNi−1−1i−1yjsmNi−1−1i−1,tNi−1−1i−1−pjsmNi−1−1i−1,tNi−1−1i−1+βji−1mNi−1−1i−1,tNi−1−1i−1+∑l≠jnγj,li−1mNi−1−1i−1,tNi−1−1i−1plsmNi−1−1i−1,tNi−1−1i−1Δt+σj,ji−1mNi−1−1i−1,tNi−1−1i−1pjsmNi−1−1i−1,tNi−1−1i−1ΔZj,jmNi−1i−1,tNi−1i−1+pjsmNi−1−1i−1,tNi−1−1i−1∑l≠jnσj,li−1mNi−1−1i−1,tNi−1−1i−1plsmNi−1−1i−1,tNi−1−1i−1ΔZj,lmNi−1i−1,tNi−1i−1.From this, we calculateπ^ji and θ^ji as:
(84)π^ji=Eyji−1TiFTi−1i−1yjsmNi−1i−1,tNi−1i−1θ^ji=Epji−1TiFTi−1i−1pjsmNi−1i−1,tNi−1i−1.Thus,yji,sTi=π^jiyji−1,sTi−,Ti−1,yi−1,s and pji,sTi=θ^jipji−1,sTi−,Ti−1,yi−1,s,pi−1,s. Let (ysmki−1,tki−1mki−1∈OSki−1, psmki−1,tki−1mki−1∈OSki−1) be a mki−1- local sequence of simulated values corresponding to mki−1-admissible lagged adapted finite sequence of conditional observation belonging to ASki−1, and corresponding term of sequence.(Uk,Bk,Kk,γk,δk,σk). Thus, for each i∈I1,K∗, (ysmki−1,tki−1mki−1∈OSki−1, psmki−1,tki−1mki−1∈OSki−1) are the finite sequence correspondence of simulated values of Eyi−1tki−1Fk−1i−1,Epi−1tki−1Fk−1i−1 at tki−1.


### 7.4. Mean-Square Suboptimal Procedure

To find the best estimate ofEytki−1Fk−1i−1,Eptki−1Fk−1i−1 using a local admissible finite sequence ysmki−1,tki−1mki−1∈OSki−1,psmki−1,tki−1mki−1∈OSki−1, we need to compute a finite sequence of quadratic mean square error corresponding to ysmki−1,tki−1mki−1∈OSki−1,psmki−1,tki−1mki−1∈OSki−1. The quadratic mean square error is defined below.Definition 21.
For eachi∈I1,K∗, the quadratic mean square error of Eytki−1Fk−1i−1,Eptki−1Fk−1i−1 relative to each member of the term of local admissible sequence ysmki−1,tki−1mki−1∈OSki−1,psmki−1,tki−1mki−1∈OSki−1 of simulated values is defined by
(85)Ξmki−1,tki−1=ysmki−1,tki−1−Eyi−1tki−1Fk−1i−12+psmki−1,tki−1−Epi−1tki−1Fk−1i−12.For any arbitrary small positive numberε and for each time tki−1, to find the the best estimate from the admissible simulated values of simulated sequence of ysmki−1,tki−1mki−1∈OSki−1,psmki−1,tki−1mki−1∈OSki−1 for Eytki−1Fk−1i−1,Eptki−1Fk−1i−1, we determine the following sub-optimal admissible set of mki−1-size local conditional sample
(86)Mtki−1=mki−1∈OSki−1:Ξmki−1,tki−1<ε,fori∈I1,K∗.Among these collected values, the value that gives the minimumΞmki−1,tki−1 for k∈0,Ni−1 are recorded as m^ki−1. If more than one value exist, then the largest of such mki−1’s is recorded as m^ki−1. If condition (86) is not met at time tki−1, the value of mki−1 where the minimum minmki−1Ξmki−1,tki−1 is attained is recorded as m^ki−1. The ε− level sub-optimal estimates of the parameters u^ji−1mki−1,tki−1,κ^j,li−1mki−1,tki−1,β^ji−1mki−1,tki−1,δ^j,li−1mki−1,tki−1,γ^j,li−1mki−1,tki−1,σ^j,li−1mki−1,tki−1 are recorded as uji−1m^k,yji−1,k,κj,li−1m^k,yji−1,k,βji−1m^k,pji−1,k,δj,li−1m^k,yji−1,k,γj,li−1m^k,pji−1,k,σj,li−1m^k,pji−1,k. Finally, the simulated value ysmki−1,tki−1, psmki−1,tki−1 at time tki−1 with m^ki−1 is now recorded as the best estimate for Eyi−1tki−1∣Fk−1i−1 and Epi−1tki−1∣Fk−1i−1. The value ysm^ki−1,k, psm^ki−1,k is called the ε− sub-optimal simulated value of ysmki−1,tki−1 and psmki−1,tki−1 of Eyi−1tki−1∣Fk−1i−1 and Epi−1tki−1∣Fk−1i−1 at tki−1.


### 7.5. Illustration: Application of Conceptual Computational Algorithm to Energy Commodity Data Set

In this subsection, we apply the above conceptual computational algorithm to study the relationship between three energy commodities by settingn=3 in (56). The three energy commodities are daily Henry Hub Natural gas data set, daily crude oil data set, and daily coal data set for the period of 05/04/2009-01/03/2014, [35–37]. Thus, for each pair y1,p1, y2,p2, and y3,p3, the drift and diffusion coefficient function of the stochastic dynamic equation governing yj,pj, for j∈I1,3 has 4 and 3 parameters each to be estimated, respectively. Thus, there are 42 parameters to be estimated in total. Using Δt=1, ε=0.001, for each j∈I1,3, the ε− level sub-optimal estimates of parameters uji−1m^ki−1,k, βji−1m^ki−1,k, κj,li−1m^ki−1,k, γj,li−1m^ki−1,k, δj,li−1m^ki−1,k, σj,li−1m^ki−1,k, l∈I1,3, at each real data times are exhibited below.


#### 7.5.1. Illustration: Relationship between Natural Gas, Crude Oil, and Coal: Without Incorporating Jump Process

In this subsubsection, we analyze the relationship between natural gas, crude oil, and coal without the jump process. Forj,l∈I1,3, the stochastic dynamic system governing the three energy commodities is described in (59) of Remark 12. Here, (y1,p1) denotes the mean spot and the spot price process of natural gas, (y2,p2) denotes the mean spot and the spot price process of crude oil, and (y3,p3) denotes the mean spot and the spot price process of coal.Using the discretized scheme (82), we apply the above conceptual computational algorithm for the real-time data sets namely daily Henry Hub natural gas data set, daily crude oil data set, and daily coal data set for the period of 05/04/2009-01/03/2014, [35–37]. Using Δt=1, ε=0.001, r=10, and d=2, the ε-level suboptimal estimates of the parameters at each real data times are described below for each commodity data sets.The parameters corresponding to the natural gas data set areu1m^k,k, β1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, σ1,1m^k,k, σ1,2m^k,k, and σ1,3m^k,k. The parameters corresponding to the crude oil data set are u2m^k,k, β2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, and σ2,3m^k,k. The parameters corresponding to coal data set are u3m^k,k, β3m^k,p3,k, κ3,1m^k,k, κ3,2m^k,k, κ3,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, γ3,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, δ3,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k.The following table gives the drift coefficient’s parameter estimatesu1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for the decoupled dynamical system for y in the case where jump is not incorporated into the dynamical system.Table1 shows the estimates of the ε suboptimal size m^k, j∈I1,3, the parameters u1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for each of the energy commodity data sets. Moreover, d≤r and the initial real data time is tr=t10.Table 1
Estimatesm^k, u1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k.
tk
m^k
Natural gas
Crude oil
Coal
u1m^k,k
κ1,1m^k,k
κ1,2m^k,k
κ1,3m^k,k
u2m^k,k
κ2,1m^k,k
κ2,2m^k,k
κ2,3m^k,k
u3m^k,k
κ3,1m^k,k
κ3,2m^k,k
κ3,3m^k,k
×10−16
×10−18
×10−18
×10−18
×10−18
×10−18
11
1
4.1593
0.0211
0
0
57.7000
0
0
0
16.7407
0
0
0
12
3
4.2000
0.0111
0
0
58.6313
0.0011
0.0310
-0.0012
16.2395
0
0
-0.0376
13
5
4.0616
0.0679
-0.0054
-0.0035
58.5378
-0.0035
0.0205
0.0032
16.2680
0
0
0.1069
14
5
4.0616
-0.0242
-0.0179
0
61.4809
0.0020
0.0098
0
15.5249
0
0
-0.0294
15
8
4.0910
0.6416
-0.2898
0
58.9282
-0.0036
0.0128
0.0071
16.8286
0
0
0.0513
16
8
4.0160
0.2101
0
0
59.6867
-0.0051
0.0080
0.0071
17.0888
0
0
0.0415
17
8
4.9575
0.1876
0
0
60.6244
0.0024
0.0052
0
17.4120
-0.0003
0.0001
0.0555
18
8
4.9575
-0.1947
0
0
61.0700
0
0
0
17.2374
0
-0
0
19
6
4.7336
-1.4476
5.8820
0
61.9414
0
0.0043
-0.0086
16.8438
0.0001
0.0001
0.0768
20
6
2.5646
0.3319
0.7261
0
62.7899
0
0.0053
0.0082
18.3022
-0.0083
0.0027
0.0558
…
…
…
…
…
…
…
…
…
…
…
…
…
…
495
8
3.9654
0.0591
-0
0
108.2457
0.0038
0.0049
-0.0023
33.1313
0.0027
0.0009
0.0363
496
5
4.0421
0.0616
0.0001
0.0017
107.5186
0
0
0
33.4224
-0.0005
0.0003
0.0214
497
6
4.0514
0.0127
-0.0002
0.0020
109.8836
0
0
-0.0001
33.3388
0.0002
0
0.0443
498
7
4.1646
0.0442
-0.0012
-0.0053
107.8013
-0.0021
0.0033
0.0038
33.2862
-0.0002
0.0006
0.0343
499
6
4.1226
0.0352
-0.0020
0
108.1554
-0.0005
0.0032
0.0039
33.2862
0.0010
0.0001
0.0068
500
6
4.2625
0.0733
-0.0002
0
110.5101
-0.0032
0.0033
0.0016
36.1647
0.0003
0.0003
0.0079
501
8
3.1551
0
0
-0.0009
110.3071
0.0014
0.0025
0
34.7467
0
0
0
502
4
4.1564
0.0914
-0.0002
0
111.1186
0
0.0013
-0.0031
49.4050
0.0026
-0.0002
0.0211
503
5
4.5799
0.0467
0.0004
0
112.0057
0
0.0027
-0.0043
34.7207
-0.0001
-0.0001
0.0216
504
4
4.3061
0.0236
0.0002
0.0007
112.3186
0
0.0021
0.0015
34.4483
0.0019
0.0003
0.0170
505
9
4.4325
-0.0015
-0.0018
0.0030
106.3345
0
0.0043
0.0001
33.7160
0
-0.0006
0.0265
…
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
7
3.5429
-0.0286
-0.0006
-0.0028
110.3777
0.0006
0.0045
0
5.2399
0
0.0013
0.0008
1103
4
3.5601
0.1028
0.0001
0.0001
111.1585
-0.0003
0.0083
0
5.4824
0
0.0077
0.0485
1104
4
3.5314
0.0809
0.0018
0.0090
109.0996
-0.0007
0.0095
0.0013
11.0949
-0.0018
0.0005
0.1175
1105
4
3.4439
0.1551
-0.0008
-0.0015
106.5667
0.0033
0.0073
-0.0020
4.8300
-0.0012
-0.0003
0.1283
1106
6
3.8206
0.2258
0.0004
0
104.7497
0
0
0.0027
4.8300
0
0.0008
0
1107
4
3.6917
0.2132
-0.0001
-0.0008
105.1229
0.0011
0.0039
0
4.3586
-0.0005
0.0004
0.1418
1108
5
3.7871
0
0
0
105.3595
0.0006
0.0027
-0.0009
4.8000
0.0006
-0.0001
0.1265
1109
4
3.8445
-0.0405
-0.0011
0.0011
102.9022
-0.0044
0.0037
0.0039
5.0279
0
0
0
1110
5
3.8399
0.0212
0.0004
0
102.8313
-0.0020
0.0045
0.0018
4.6817
0.0021
0.0041
0.0536Figure1 shows the drift coefficient’s parameter estimates u1m^k,k, u2m^k,k, and u3m^k,k for the decoupled dynamical system for y in the case where jump is not incorporated into the dynamical system.Figure 1
The graph of mean levelu1m^k,k, u2m^k,k, and u3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)Figures1(a)–1(c) are the graphs of u1m^k,k, u2m^k,k, and u3m^k,k against time tk for the daily Henry Hub natural gas price [36], daily crude oil price [37], and daily coal price [35] data set, respectively. By plotting the real data sets, it is easily seen that the graphs of u1m^k,k, u2m^k,k, and u3m^k,k are similar to the graph of the Henry Hub natural gas, crude oil, and coal data set, respectively. We expect this to happen because uj, j∈I1,3, are the equilibrium spot price processes of (5). The sample means of the real spot price data sets for natural gas, crude oil, and coal are given by 3.7218, 88.5620, and 16.5543, respectively. It can be seen from Figures 1(a)–1(c) that the graphs of u1m^k,k, u2m^k,k, and u3m^k,k move around the mean values 3.7218, 88.5620, and 16.5543, respectively. This analysis shows that the parameters uj, j∈I1,3, are statistic processes for the respective mean of the data sets at time tk.The graph of the interaction parametersκ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for the decoupled dynamical system for y (with no jump incorporated into the dynamical system) are given in Figure 2 below.Figure 2
The graph of interaction coefficientsκ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures2(a)–2(i) show the graph of the ε suboptimal interaction coefficient parameters κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k. The interaction coefficients κj,l, j≠l are negligible, because each estimate is <<10−15. Thus, this shows that the model describing the mean spot price, yj, is mainly characterized by the market potential κj,juj−yjyj, j∈I1,n.Table2 below shows the estimates of the diffusion coefficient’s parameters δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k,δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k for the decoupled dynamical system for y in the case where jump is not incorporated into the dynamical system.Table 2
Estimatesδ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k,δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
δ1,1m^k,k
δ1,2m^k,k
δ1,3m^k,k
δ2,1m^k,k
δ2,2m^k,k
δ2,3m^k,k
δ3,1m^k,k
δ3,2m^k,k
δ3,3m^k,k
11
0.0123
0.0012
0.0001
0
0.0223
0
0.0412
0
0.0022
12
0.0024
0.0011
0.0121
0.0234
0.0245
0
0
0
0.0112
13
0.0001
1.3425
1.7280
1.9811
0.9899
0.9731
0.6374
0.6374
0.0123
14
0
1.1267
0.6027
2.3258
0.1213
3.9128
1.6564
1.6564
0.0004
15
1.15260
0.4287
0.6210
2.3252
0.0006
0.5083
1.6650
1.6650
0.4565
16
4.9354
0
0
2.3217
0.0120
1.1124
1.6724
1.6724
0.8762
17
4.1360
0.0989
3.6877
1.6425
0
0
1.7719
1.7719
0
18
3.0410
0.1527
0
1.3105
0.9167
1.3451
1.7630
1.7630
0
19
2.7713
0
0
1.1052
0
3.3241
1.7400
1.7400
0
20
2.8461
0.0012
0.2221
0.1196
5.1929
0
0.6532
0.9876
0.0082
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
494
2.9961
0.0586
0
0.5529
0
0.42339
0
0
0.5187
495
5.9059
0
0.0584
0.5488
0.8947
0
0.0017
0.0021
0.0001
496
0.1121
0
0.6613
0.5767
0.9899
0
0.8763
0
0.9827
497
1.1229
0.0095
0.0988
0.6499
5.8547
0
1.1317
1.1317
0.0012
498
0.6946
0.0101
0
0
5.8298
0.0320
1.0294
1.0294
0.0321
499
0.7353
0.0066
0.0384
0
5.7180
0.0330
0.7317
0.7317
0.0431
500
1.7509
0.0069
0.0283
0.4307
5.6133
0.0413
0.4826
0.4826
0.0783
501
2.1299
0.0077
0.0282
0.5043
5.6282
0.0308
0.4272
0.4272
0.0002
502
0.9778
0.0077
0.0255
0.2878
4.6543
0.0322
0.5239
0.5239
0.0098
503
0.9872
0
0
0.2909
4.5544
0.0411
1.4523
1.4523
0.0087
504
1.1329
0
0
0.3707
0
0.1128
2.4181
2.4181
0
505
1.9178
0
0
0.3812
1.3243
0.1724
4.9207
4.9207
0
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
0
0.0331
0.0056
0.9297
3.9502
0
0.2853
1.8033
1.1355
1103
1.5077
0.0626
0.0332
1.1017
2.8221
0
0
0
1.4133
1104
0
0.0435
0.5821
0.1939
4.5585
0
0
0
1.1672
1105
0
0
1.52970
0.1922
3.2418
0.7273
0.2726
0.2726
0
1106
4.4476
0.323
0.5112
3.5487
3.8113
1.0179
0.3296
0.3296
0
1107
2.4312
0.0011
0.0435
0.2001
2.6026
0.9354
0
0
1.7245
1108
2.5079
0.1232
0.4542
0.3781
0
0.8825
0.1878
0.1878
0
1109
1.7828
0.0431
0.3210
0.4024
0
0.8812
0
0
1.3191
1110
1.2706
0.0056
1.1123
0.3252
0
0.8078
0
0
1.0233The graph of the diffusion coefficient’s parameter for the decoupled dynamical system fory without jump incorporated into the dynamical system are given in Figure 3 below.Figure 3
The graph of interaction coefficientsδ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures3(a)–3(i) show the graphs of the ε suboptimal interaction measure of fluctuation coefficient parameters δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.Table3 gives the drift coefficient’s parameter estimates β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k for the dynamical system for p without jump.Table 3
Estimatesm^k, β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, m^k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, m^k,β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
β1m^k,k
γ1,1m^k,k
γ1,2m^k,k
γ1,3m^k,k
β2m^k,k
γ2,1m^k,k
γ2,2m^k,k
γ2,3m^k,k
β3m^k,k
γ3,1m^k,k
γ3,2m^k,k
γ3,3m^k,k
11
0.3462
0.2579
-0.0039
0.0218
0.0029
0.0023
0.0056
0.0124
0
0
0.0432
0.0012
12
0.1681
0.3497
-0.0109
0.0248
0
0
0
0
0.6812
-0.3513
0.0248
0
13
0.1592
0.3755
-0.0102
0.0228
-0.0490
0.0228
0
-0.0027
1.0795
-0.2904
0.0135
0
14
-0.0889
0.5001
-0.0069
0.0257
-0.3700
-0.1689
0
0.0633
-0.6689
-0.1969
0.0166
0.0177
15
0.7025
0.6513
-0.0242
0.0376
-0.0903
-0.1737
0
0.0477
-0.8220
-0.0944
0.0061
0.0291
16
0.6727
0.6513
0.0188
-0.1048
0.3425
-0.1847
0
0.0252
-0.0480
0.0694
-0.0112
0.0165
17
0.3253
0.3674
-0.0103
0.0140
-0.4531
-0.1457
0
0.0638
-3.6240
0.1086
0.0266
0.0569
18
0.1523
0.3433
0.0014
-0.0163
4.0859
0.2320
0.3114
-0.2889
-1.3277
-0.0566
0.0088
0.0353
19
-7.9573
0.3433
-0.2058
1.1496
0.1389
-0.0677
0.0004
0.0065
-0.9232
0.0368
-0.0277
0.0883
20
0.0514
0.3028
0.0041
-0.0192
-0.1464
-0.0409
0.0004
0.0178
-1.9138
0.0530
-0.0104
0.0836
…
…
…
…
…
…
…
…
…
…
…
…
…
495
0.0552
0.2994
0.0015
-0.0063
1.8435
-0.1608
0.1040
-0.0368
0.5815
-0.3597
-0.0050
0.0350
496
0.1799
0.1860
0.0004
-0.0067
1.5682
-0.0268
0.4723
-0.0502
0.8586
-0.2029
-0.0094
0.0241
497
0.8047
0.1923
0.0023
-0.0314
6.6699
-0.2655
0.4723
-0.1649
0.7191
-0.2061
-0.0053
0.0172
498
0.2742
0.2651
0.0020
-0.0145
1.0042
0.0088
0.0226
-0.0315
0.3978
-0.1680
-0.0034
0.0161
499
0.4915
0.2295
-0.0006
-0.0125
1.3074
0.3761
0.0073
-0.0872
0.1425
-0.1899
-0.0026
0.0224
500
0.5659
0.1618
0.0008
-0.0194
0.4040
-0.0889
0.0392
-0.0011
0.3674
-0.2313
-0.0073
0.0331
501
0.4498
0.1679
0.0010
-0.0167
0.4230
-0.1297
0.0434
0.0035
-0.9002
1.1703
-0.0775
0.1011
502
0.4836
0.1850
-0.0001
-0.0139
0.5570
-0.1502
0.0384
0.0022
-0.2313
0.6524
-0.0496
0.0663
503
0.4696
0.1850
0.1224
-0.0919
-0.0441
0.0299
0.0384
-0.0023
3.7804
0.0120
-0.0498
0.0389
504
-0.0456
0.1850
0.0088
-0.0270
0.6112
-0.0820
0.0425
-0.0080
6.4696
0.4005
-0.0950
0.0543
505
0.0464
1.7125
-0.0423
0.1339
0.7135
-0.1115
0.1135
-0.0082
2.2295
0.0897
-0.0357
0.0306
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
0.6765
0.0455
-0.0020
-0.0908
0.2863
-0.2183
0.1891
0.1028
4.3927
3.8144
0.1072
0.0250
1103
1.1804
0.4214
-0.0149
0.0837
-2.1858
0.4491
0.1891
0.1135
-6.1960
0.7446
0.0261
0.0144
1104
0.1069
0.2489
-0.0009
-0.0014
-2.1178
0.3406
0.1959
0.1826
-6.4415
0.0339
-0.0037
0.1429
1105
0.2367
0.0128
0.0065
-0.0019
0.2633
0.0565
0.0742
-0.0997
0.6510
2.4930
0.0714
0.1429
1106
0.1178
0
0
-0.0014
0.1384
0.0784
0.2014
-0.0904
0.6510
2.4930
0.6341
0.1429
1107
0.1466
0.4648
-0.0002
-0.0271
0.5787
-0.0297
0.1305
-0.0979
-4.5897
-0.0971
0.0240
0.0509
1108
0.3240
0.2478
0.1212
-0.0074
0.4293
-0.0678
0.0721
-0.0389
-4.5961
-0.0734
0.0233
0.0507
1109
0.121
0
0
-0.0021
0.2282
-0.0468
0.0721
-0.0133
-4.5961
-2.6776
0.5625
0.4702
1110
0.002
0
0
-0.0056
0.1523
0.0121
0.0011
-0.0129
9.5959
0.9045
-0.1478
0.0499Table3 shows the estimates of the ε suboptimal size m^k, j∈I1,3 and the parameters β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k, for each of the energy commodity data sets. Moreover, d≤r, and the initial real data time is tr=t10.In the following, we give the graph of the drift coefficient’s parameters with estimates in Table3 in Figure 4 below.Figure 4
The graph of interaction coefficientsγ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures4(a)–4(i) show the graphs of the ε suboptimal interaction coefficient parameters γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k. According to (58), the estimate γj,lm^k,k, j≠l, is positive if commodity pl is cooperating with commodity pj, and negative if commodity pl is competing with commodity pj. There is no interaction between the two commodities if γj,lm^k,k=0. It is apparent from the graph of γ1,3m^k,k that coal is competing with natural gas because the estimates of γ1,3m^k,k are mostly negative. Also, it is apparent that natural gas and crude oil are either cooperating or competing, depending on the time period.Figures5(a)–5(c) are the graphs of β1m^k,k, β2m^k,k, and β3m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively.Figure 5
The graph ofβ1m^k,k, β2m^k,k, and β3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)Table4 gives the ε suboptimal estimates of the parameters σ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k for each of the energy commodity data sets.Table 4
Estimatesσ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
σ1,1m^k,k
σ1,2m^k,k
σ1,3m^k,k
σ2,1m^k,k
σ2,2m^k,k
σ2,3m^k,k
σ3,1m^k,k
σ3,2m^k,k
σ3,3m^k,k
11
0
0
0
0
0
0
0
0
0
12
0.0485
0.0004
0.0032
0.2734
0.0166
0
0
0
0.0000
13
0
0
0
0
0
0
0
0
0
14
0.2120
0.1386
0.0133
1.2573
0.4773
0.1195
0
0.0665
0.0086
15
0.4246
0.1318
0.0021
2.1081
0.4894
0.1211
0
0.6107
0.0696
16
0.5538
0.0778
0.1501
0
0.2524
0.0811
0.0651
0.4251
0.0635
17
1.1121
0.0469
0.2230
0
0.1848
0.2463
0
0.4458
0.0478
18
1.5347
0.0180
0.2178
0
0.1877
0.1602
0.5681
0.0592
0.0115
19
1.1315
0.0619
0.2221
0
0.2673
0.2465
0.4999
0.0569
0.0127
20
2.0845
0.0536
0.1866
0
0.1700
0.0781
0.3789
0.3174
0.0046
…
…
…
…
…
…
…
…
…
…
495
0
0.0036
0.0406
0.2286
0.0600
0.0172
0
0.9387
0.0182
496
0.1588
0.0035
0.0107
1.4847
0.3163
0.0102
0
0
0.0016
497
0.1551
0.0009
0.0065
0
0.1453
0
0.7777
0
0.0033
498
0.1576
0.0011
0.0073
0
0.1679
0
0.5334
0
0.0060
499
0.1197
0.0006
0.0059
1.9414
0.2391
0.0172
0.4405
0.1432
0.0097
500
0.3600
0.0001
0.0049
1.9554
0.3960
0.0079
0.6331
0.1410
0.0093
501
0.0514
0.0033
0.0049
2.0436
0.3499
0.0111
0.7690
0.1376
0.0089
502
0.2503
0.0034
0.0042
2.0837
0.1744
0.0132
0.6198
0.1274
0.0066
503
0.1195
0.0147
0.0165
0
0.4283
0.0060
1.1613
0.1530
0.0049
504
0.0974
0.0144
0.0027
0
0.2241
0.0048
0.4778
0.0574
0.0043
505
0.1422
0.0060
0.0131
0
0.2023
0.0054
0.5604
0.0669
0.0004
…
…
…
…
…
…
…
…
…
…
1102
0.1898
0.0016
0.0413
0.8313
0.0767
0.0381
0.6875
0
0.1451
1103
0.2094
0.0015
0.0352
0.8262
0.0673
0.0451
0.7298
0.2808
0.0147
1104
0.1711
0.0011
0.0040
0.6648
0.0915
0.0462
0.5563
0.1831
0.0105
1105
0.1816
0.0012
0.0116
0.6658
0.1049
0.0371
0.6591
0.2874
0.0057
1106
0.1191
0.0011
0.0116
0.6260
0.1155
0.0393
0
0.0196
0.0060
1107
0.0417
0.0012
0.0041
0.4992
0.0781
0.0382
0
0
0.0065
1108
0.1058
0.0033
0.0045
0.0019
0.0589
0.0421
0
0
0.0018
1109
0.1740
0.0021
0
0
0.0446
0.0316
2.1187
0
0.4511
1110
0.2912
0.0021
0.0163
0.0385
0.0342
0.0037
0
1.1563
0.0257Figures6(a)–6(i) are the graphs of σ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ1,1m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively.Figure 6
The graph ofσ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ1,1m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Table5 shows the real and simulated estimates for the spot price processes pjt, j∈I1,n.Table 5
Real and simulated estimates (without jump) for natural gas, crude oil, and coal.
tk
Real
Natural gas
Real
Crude oil
Real
Coal
Simulatedp1sm^k,k
Simulatedp2sm^k,k
Simulatedp3sm^k,k
11
4.0200
4.0500
58.9900
56.5200
16.5900
16.8000
12
3.9900
4.0500
59.5200
59.3099
17.4600
16.8635
13
3.7500
3.6690
61.4500
59.3377
17.8900
17.8086
14
3.7700
3.6341
60.4900
59.4191
17.5500
17.0859
15
3.4100
3.3967
61.1500
59.6974
17.4100
17.0859
16
3.3500
3.3967
62.4800
59.6974
16.7500
17.0859
17
3.4900
3.4537
63.4100
61.2177
17.6600
19.0677
18
3.5500
3.4537
65.0900
61.4561
17.5200
16.0578
19
3.9200
3.8618
66.3100
61.6529
18.5000
19.0677
20
3.8600
3.8618
68.5900
60.9364
19.0600
19.0677
…
…
…
…
…
…
…
495
4.1900
4.0368
107.1800
104.1295
32.7600
31.3108
496
4.3300
4.1868
110.8400
111.1245
33.6500
32.7737
497
4.3300
4.1025
111.7200
112.4675
33.7100
33.4888
498
4.3700
4.0964
111.6800
110.8795
34.7500
35.5907
499
4.3200
4.1042
111.7200
104.2465
34.5400
32.9391
500
4.3500
4.0548
112.3100
109.9535
34.0400
36.2674
501
4.3800
4.0548
112.3800
109.9995
33.1000
36.2674
502
4.5100
4.3249
113.3900
104.3254
33.6700
34.8915
503
4.6000
4.3555
113.0300
113.2356
33.9400
35.0472
504
4.6000
4.3491
110.6000
103.9435
33.8300
32.8992
505
4.5900
4.3609
108.7900
104.9995
32.0200
32.8992
…
…
…
…
…
…
…
1102
3.7200
3.5963
108.2300
110.5149
4.7700
2.8861
1103
3.7300
3.5963
106.2600
105.8076
5.0100
5.6871
1104
3.6800
3.4099
104.7000
105.8076
4.9800
5.3821
1105
3.6600
3.4356
103.6200
105.8076
4.7300
4.9221
1106
3.5900
3.4636
103.2200
106.9547
4.6800
4.2352
1107
3.5200
3.2573
102.6800
105.4047
4.6300
5.8172
1108
3.4900
2.8981
103.1000
102.4928
4.7400
6.0376
1109
3.5100
2.8981
102.8600
102.4928
4.3300
5.1121
1110
3.4800
3.0267
102.3600
102.4928
4.1800
4.8978Figure7 below shows the graph of the real and simulated prices for natural gas, crude oil, and coal data set.Figure 7
Real and simulated prices (without jump) for natural gas, crude oil, and coal.
(a)
(b)
(c)Figures7(a)–7(c) show the graph of the real and simulated spot prices for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively. The red line represents the real data set ptk, while the blue line represent the simulated data set psm^k,k. Here, we begin by using a starting delay of r=10. The simulation starts from tr=t10. It is clear that the graph fits well. To reduce magnitude of error, we increase the magnitude of time delay. We later compare this result with the case where jump is incorporated into the system.


#### 7.5.2. Relationship between Natural Gas, Crude Oil, and Coal: With Jump Incorporated

In this subsubsection, we analyze the relationship between natural gas, crude oil, and coal with the jump process. Here, we apply the above conceptual computational algorithm described in this section for the real-time data sets, namely, daily Henry Hub natural gas data set, daily crude oil data set, and daily coal data set for the period of 05/04/2009-01/03/2014, [35–37]. For i∈I1,K∗, K≠0, we use Δti−1=1, ε=0.001, ri−1=10, and di−1=2. The ε-level suboptimal estimates of the parameters at each real data times are described below for each commodity data sets.The parameters corresponding to the model governing natural gas price data set areu1i−1m^k,k, β1i−1m^ki−1,tki−1, κ1,1i−1m^ki−1,tki−1, κ1,2i−1m^ki−1,tki−1, κ1,3i−1m^ki−1,tki−1, γ1,1i−1m^ki−1,tki−1, γ1,2i−1m^ki−1,tki−1, γ1,3i−1m^ki−1,tki−1, δ1,1i−1m^ki−1,tki−1, δ1,2i−1m^ki−1,tki−1, δ1,3i−1m^ki−1,tki−1, σ1,1i−1m^ki−1,tki−1, σ1,2i−1m^ki−1,tki−1, and σ1,3i−1m^ki−1,tki−1. The parameters corresponding to the model governing crude oil price data set are u2i−1m^ki−1,tki−1, β2i−1m^ki−1,tki−1, κ2,1i−1m^ki−1,tki−1, κ2,2i−1m^ki−1,tki−1, κ2,3i−1m^ki−1,tki−1, γ2,1i−1m^ki−1,tki−1, γ2,2i−1m^ki−1,tki−1, γ2,3i−1m^ki−1,tki−1, δ2,1i−1m^ki−1,tki−1, δ2,2i−1m^ki−1,tki−1, δ2,3i−1m^ki−1,tki−1, σ2,1i−1m^ki−1,tki−1, σ2,2i−1m^ki−1,tki−1, and σ2,3i−1m^ki−1,tki−1, while the parameters corresponding to the model governing coal price data set are u3i−1m^ki−1,tki−1, β3i−1m^ki−1,tki−1, κ3,1i−1m^ki−1,tki−1, κ3,2i−1m^ki−1,tki−1, κ3,3i−1m^ki−1,tki−1, γ3,1i−1m^ki−1,tki−1, γ3,2i−1m^ki−1,tki−1, γ3,3i−1m^ki−1,tki−1, δ3,1i−1m^ki−1,tki−1, δ3,2i−1m^ki−1,tki−1, δ3,3i−1m^ki−1,tki−1, σ3,1i−1m^ki−1,tki−1, σ3,2i−1m^ki−1,tki−1, and σ3,3i−1m^ki−1,tki−1.For the sake of simplicity and in order to be able to compare our results in this subsection with the results in Subsection7.5.1, for each j,l∈I1,n, we rewrite the parameters uji−1m^ki−1,tki−1, βji−1m^ki−1,tki−1, κj,li−1m^ki−1,tki−1, γj,li−1m^ki−1,tki−1, δj,li−1m^ki−1,tki−1, and σj,li−1m^ki−1,tki−1 after they have been estimated as ujm^k,k, βjm^k,k, κj,lm^k,k, γj,lm^k,k, δj,lm^k,k, and σj,lm^k,k.We give some results for the jump times of the systemTii∈I1,K∗.Table6 shows some result for the jump times of the system (y,p). These results are derived by recording the times at which an entry of a commodity differ by twice the standard deviation or more from the mean of that commodity. These times are now combined into a single array and sorted out in an increasing order.Table 6
Some result for the jump times of the system (y,p).
T
17
44
61
87
157
200
422
464
483
502
722
754
870
930
1113We give the estimate of the jump coefficient matricesΠi and Θi defined in (34) in Table 7 below.Table 7
Estimatesπ1i, π2i, π3i, θ1i, θ2i, and θ3i.
i
Ti
Natural gas
Crude oil
Coal
Ti
Natural gas
Crude oil
Coal
Ti
Π1i
Π2i
Π3i
Ti
θ1i
θ2i
θ3i
1
17
1.0031
1.1219
1.0256
17
1.0049
1.1219
1.0493
2
44
0.9213
0.9727
1.0410
44
0.9352
1.0084
0.9249
3
61
0.9482
0.9671
0.9661
61
0.9997
0.9427
0.9404
4
87
0.8859
0.9974
0.9653
87
0.7389
1.0452
0.9905
5
157
1.0435
0.9350
1.0432
157
1.0933
1.0019
1.0049
6
200
1.0309
1.0199
1.0382
200
0.9826
1.0210
0.9794
7
422
1.0270
0.9775
0.9669
422
0.9706
0.9939
0.9917
8
464
0.9581
1.0462
1.0523
464
1.0128
1.0508
1.0324
9
483
0.9765
0.9787
1.0291
483
1.0382
1.0328
1.0246
10
502
1.0532
1.0737
1.0136
502
1.0359
1.0073
1.0162
11
722
0.9812
0.9959
0.9919
722
0.9700
0.9695
1.0011
12
754
1.0003
1.0009
0.9189
754
1.0137
0.9987
1.3481
13
870
1.0579
0.9921
1.1378
870
1.0328
1.0033
1.1420
14
930
1.0275
0.9907
0.9978
930
0.9995
0.9812
1.1848
15
1113
1.0009
0.9960
1.0706
1113
0.9304
0.9801
0.9897Table8 gives the estimates u1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for the decoupled dynamical system for y with jump.Table 8
Estimatesm^k, u1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, m^k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, m^k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k.
tk
m^k
Natural gas
Crude oil
Coal
u1m^k,k
κ1,1m^k,k
κ1,2m^k,k
κ1,3m^k,k
u2m^k,k
κ2,1m^k,k
κ2,2m^k,k
κ2,3m^k,k
u3m^k,k
κ3,1m^k,k
κ3,2m^k,k
κ3,3m^k,k
×10−16
×10−16
×10−16
×10−16
×10−16
×10−16
11
1
4.1593
0
0
0
57.7000
0
0
0
16.7407
0
0
0
12
4
4.2000
0
0
0
58.6313
0.0011
0.0310
-0.0012
16.2395
0
0
-0.0376
13
6
4.0616
0.0679
-0.0054
-0.0035
58.5378
-0.0035
0.0205
0.0032
16.2680
0
0
0.1069
14
2
4.0616
-0.0242
-0.0179
-0.0035
61.4809
0.0020
0.0098
0
15.5249
0
0
-0.0294
15
7
4.0910
0.6416
-0.2898
0.0078
22.5758
-0.0057
0.0085
0.0012
18.7073
0.0009
-0.0021
0.0318
16
4
4.0160
0
0
0.0078
59.6867
-0.0051
0.0080
0
17.0060
0
-0.0021
0
17
2
4.9575
0
0
0.0078
60.3710
-0.0005
0.0207
0
12.8918
-0.0005
-0.0002
0.0318
18
8
4.9575
-0.1947
0
0.0078
62.3437
0.0005
-0.0008
0
16.5954
0.0002
0.0008
0.0662
19
4
3.3190
-0.4472
0.6760
0.0078
74.6911
-0.0008
-0.0019
0
17.9932
0
-0.0002
0
20
1
3.4762
-0.2540
-0.0048
0.0078
65.9190
0.0026
-0.0006
1
17.6485
0
0
0.0499
…
…
…
…
…
…
…
…
…
…
…
…
…
…
494
1
4.1457
0
0
-0.0001
115.1875
0.0002
0.0053
0
33.3359
0.0003
-0.0003
0.0326
495
1
4.2877
0.1184
-0.0014
0
124.5218
0.0008
0.0056
0
30.1732
0.0002
-0
0.0412
496
1
4.2238
0.2582
0.0011
0.0003
106.8349
0.0003
0.0113
-0.0006
34.9907
0.0034
0
0.0097
497
5
4.0998
0.0477
-0.0006
-0.0002
108.4725
-0.0003
0.0162
-0.0033
33.3388
0.0002
0
0.0443
498
8
4.0592
0.0201
0.0010
0
104.8926
0
0
0.0003
35.1174
0
0.0001
0.0207
499
1
4.3433
0.2118
-0.0014
0
109.2551
-0.0002
0.0048
0.0003
33.2862
0.0010
0.0001
0.0068
500
4
2.4519
0
0
0
111.7067
0
0
0
36.1647
0.0003
0.0003
0.0079
501
1
4.2415
0.4108
0
-0.0015
110.7517
-0.0006
0.0026
0.0009
34.9145
-0.0001
-0.0004
0.0407
502
2
4.3633
0.3210
0.0002
-0.0001
103.6326
-0.0019
0.0023
-0.0002
34.8337
-0.0001
-0.0001
0.0140
503
2
4.2911
0.1276
0.0003
0.0043
112.1547
-0.0033
0.0030
0.0027
35.8389
0.0005
0.0001
0.0211
504
7
4.5942
-0.0125
-0.0002
-0.0031
111.1278
0.0010
0.0072
0.0006
33.6875
-0.0021
0
0.0268
505
2
3.1882
0.0666
0.0009
0
106.1919
-0.0009
0.0110
0.0011
33.6640
-0.0011
-0.0002
0.0231
…
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
1
3.5909
0
0
0.0008
110.3777
0.0006
0.0045
0
5.1761
0.0067
-0.0029
-0.0044
1103
6
3.5303
0.1166
0.0002
-0.0003
111.1585
-0.0003
0.0083
0
5.4558
-0.0019
0.0014
0.0600
1104
4
3.5314
0.0809
0.0018
0
109.0996
-0.0007
0.0095
0.0013
4.8000
0.0005
0.0006
0.1742
1105
1
3.7100
0.2234
-0.0013
-0.0015
106.5667
0.0033
0.0073
-0.0020
5.4226
-0.0082
0.0020
0.0932
1106
8
3.4084
0.1098
0.0001
0
106.5989
0.0003
0.0030
0.0023
5.3360
-0.0023
0.0005
0.0956
1107
7
3.5520
0.1086
0.0001
-0.0070
103.4473
-0.0020
0.0037
-0.0045
4.3586
-0.0005
0.0004
0.1418
1108
5
3.9233
0.0601
0.0007
0
102.8550
0
0.0040
0
4.6582
-0.0010
0
0.1388
1109
8
3.5328
0.0417
0
0
103
-0.0002
0.0089
-0.0005
4.9663
-0.0019
0.0008
0.1279
1110
5
3.8399
0.0212
0.0004
0
102.8800
0
0
0
4.7286
-0.0037
-0.0030
0.0740Figure8 shows the parameter estimates u1m^k,k, u2m^k,k, and u3m^k,k for the decoupled dynamical system for y with jump.Figure 8
The graph of mean levelu1m^k,k, u2m^k,k, and u3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)Figures8(a)–8(c) are the graphs of u1m^k,k, u2m^k,k, and u3m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively. By plotting the real data sets, it is easily seen that the graphs of u1m^k,k, u2m^k,k, and u3m^k,k are similar to the graph of the Henry Hub natural gas, crude oil, and coal data set, respectively. We expect this to happen because uj, j∈I1,3, are the equilibrium spot price processes of (5).The graph of the interaction parametersκ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for the decoupled dynamical system for y with jump and estimates in Table 8 are given in Figure 9 below.Figure 9
The graph of interaction coefficientsκ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures9(a)–9(i)) show the graph of the ε suboptimal interaction coefficient parameters κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k. The interaction coefficients κj,l, j≠l, are negligible, because each estimate is <<10−15. Thus, this shows that the model describing the mean spot price, yj, is mainly characterized by the market potential κj,ji−1uji−1−yjyj, j∈I1,n, i∈I1,K∗.Table9 below shows the estimates of the diffussion coefficient’s parameters δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k,δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k for the decoupled dynamical system for y with jump.Table 9
Estimatesδ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
δ1,1m^k,k
δ1,2m^k,k
δ1,3m^k,k
δ2,1m^k,k
δ2,2m^k,k
δ2,3m^k,k
δ3,1m^k,k
δ3,2m^k,k
δ3,3m^k,k
11
0.0062
0.0010
0.0001
1.5277
0.0078
0.0011
0
0
0.0218
12
0.0182
0.9002
0
1.6227
0.0010
0
0
0
0.0988
13
0.0239
0.0802
1.7280
0
1.7694
0
0.6374
0.6374
0.0959
14
0
0.0001
0.6027
2.3258
0
0
1.6564
1.6564
0.0847
15
0
0.8001
0.6210
2.3252
0
0
1.6650
1.6650
0.0111
16
0.0455
0.0007
3.6877
2.3217
0
1.2215
1.6724
1.6724
0
17
0
0.9876
0
1.6425
0
0
1.7719
1.7719
0
18
3.0410
0.9351
0
1.3105
0
0.1070
1.7630
1.7630
0.0434
19
2.7713
0.6680
0
1.1052
0
0
1.7400
1.7400
0
20
2.8461
1.7795
0
0.1196
0
0.0983
0
0.4555
0
…
…
…
…
…
…
…
…
…
…
495
1.1229
0
0.0584
0.5488
0.1104
0.0761
0
0
1.3987
496
0.6946
0
0.6613
0.5767
0.0715
0.0610
0
0
1.3017
497
1.1229
0.0095
0.0988
0.6499
0.0870
0.0633
1.1317
1.1317
1.3069
498
0.6946
0.0101
0
0
0
0.0320
1.0294
1.0294
1.5410
499
0.7353
0.0066
0.0384
0
0.0922
0.0330
0.7317
0.7317
1.2225
500
1.7509
0.0069
0.0283
0.4307
0.4545
0.0413
0.4826
0.4826
1.2254
501
2.1299
0.0077
0.0282
0.5043
0.7873
0.0308
0.4272
0.4272
1.5587
502
0.9778
0.0077
0
0.2878
0
0
0.5239
0.5239
1.8713
503
0.9872
0
0
0.2909
0
0
1.4523
1.4523
1.8874
504
1.1329
0
0
0.3707
0.4261
0
0
0
0
505
1.9178
0
0
0.3812
0.7292
0.1724
0
0
0
…
…
…
…
…
…
…
…
…
…
1102
0
0.0331
0.7183
0.9297
0.0434
0.0680
0
0
1.1355
1103
1.5077
0.0626
0.2048
1.1017
0.0421
0.1510
0
0
1.4133
1104
0.4444
0.0435
0.4622
0.1939
0.1078
0
0.0814
0
1.1672
1105
3.5933
0
0.3646
0.1922
0
0.7273
0.2726
0.2726
1.3023
1106
2.4964
0
0.3919
0
0.0684
1.0179
0.3296
0.3296
1.4111
1107
2.4600
0
0.8995
0.2001
0.1510
0.9354
0
0
1.7245
1108
2.0262
0
0.6325
0.3781
0.0814
0.8825
0.1878
0.1878
1.0915
1109
1.7828
0
0.6116
0.4024
0.0332
0.8812
0
0
1.3191
1110
1.2706
0
0.1001
0.3252
0.0155
0.8078
0
0
1.0233The graph of the diffusion coefficient’s parameter for the decoupled dynamical system fory with jump are given in Figure 10 below.Figure 10
The graph of interaction coefficientsδ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures10(a)–10(i) show the graph of the ε- sub-optimal interaction measure of fluctuation coefficient parameters δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.Table10 gives the drift coefficient’s parameter estimates β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k for the dynamical system for p with jump.Table 10
Estimatesβ1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k.
tK
Natural gas
Crude oil
Coal
β1m^k,k
γ1,1m^k,k
γ1,2m^k,k
γ1,3m^k,k
β2m^k,k
γ2,1m^k,k
γ2,2m^k,k
γ2,3m^k,k
β3m^k,k
γ3,1m^k,k
γ3,2m^k,k
γ3,3m^k,k
11
0
0
0
0
0
0
0
0
0
0
0
0
12
0.1681
0.3497
-0.0109
0.0248
-0.4815
0.1626
-0.4066
-0.0123
0.7665
-0.3259
0.0205
-0.0198
13
0.1592
0.3755
-0.0102
0.0228
-0.7778
-0.5752
-0.0578
0.1870
1.0795
-0.2904
0.0135
-0.0217
14
7.3439
0.3755
0.0488
-0.5478
1.7680
-0.5555
-0.2058
0.0291
0.8543
-0.2056
0.0034
-0.0064
15
0.3336
0.3652
-0.0127
0.0213
-0.8999
0.0601
-0.1110
0.0405
0.5144
-0.1264
-0.0062
0.0127
16
0.4709
0.2780
-0.0116
0.0104
-0.8999
0.0601
-0.1110
-0.0292
0.0017
-0.0002
0
0
17
0.3277
0.2780
0.0768
-0.2633
1.3349
0.0027
-0.0330
-0.0750
-0.6285
-0.0569
0.0016
0.0262
18
0.3277
1.3156
-0.1491
-0.1646
1.5419
-0.0088
0.1205
-0.0892
-1.3275
-0.0703
0.0080
0.0386
19
0
0
0
0
-0.1785
-0.0368
-0.0062
0.0189
-0.8091
-0.0001
-0.0083
0.0466
20
0.6985
0.4990
-0.0069
-0.0187
-0.1513
-0.0778
-0.0096
0.0255
-0.0182
0.0080
0.0001
0.0003
…
…
…
…
…
…
…
…
…
…
…
…
…
495
0.1201
0.2264
0.0007
-0.0056
0.2756
0.1131
-0.9587
-0.0087
-0.0288
-0.0780
-0.0019
0.0136
496
0.1809
0.2085
0.0009
-0.0082
0.2898
-0.0431
0.7329
-0.0133
0.1324
-0.1434
-0.0018
0.0158
497
0.2442
0.1597
0.0007
-0.0093
3.1030
-0.0495
-0.0462
-0.0862
0.9772
-0.2426
-0.0065
0.0177
498
0.2742
0.2651
0.0020
-0.0145
1.3147
0.0148
-0.0009
-0.0411
0.2770
-0.1888
-0.0037
0.0217
499
0.3320
0.3298
-0.0009
-0.0070
0.8430
0.0070
0.0283
-0.0265
0.0931
-0.1551
-0.0041
0.0237
500
0.5035
0.2337
-0.0007
-0.0128
1.3320
0.3949
-0.3308
-0.0838
0.4175
-0.2331
-0.0034
0.0222
501
0.6328
0.2612
-0.0034
-0.0077
0.3251
-0.1148
0.0548
0.0042
0.7896
-0.2546
-0.0093
0.0305
502
0.5403
0.2457
-0.0014
-0.0113
0.4863
-0.1315
0.0343
0.0020
3.7990
0.0420
-0.0524
0.0421
503
0.4794
0.2098
-0.0028
-0.0050
0.1239
0.0013
0.0202
-0.0040
9.6735
0.0736
-0.1152
0.0648
504
-0.3308
-0.5600
0.0258
-0.0737
0.2867
-0.0391
0.0009
-0.0034
4.2547
0.3506
-0.0669
0.0382
505
1.1680
0.8346
-0.0274
0.0542
0.1198
-0.0588
-0.3412
0.0092
2.2295
0.0897
-0.0357
0.0306
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
0.6765
0.0455
-0.0020
-0.0908
0.4026
-0.2544
0.2045
0.1058
-5.9294
0.6777
0.0292
0.0068
1103
1.1804
0.4214
-0.0149
0.0837
-0.6549
0.1780
0.0070
0.0018
-6.3380
0.7440
0.0291
0.0106
1104
0.1069
0.2489
-0.0009
-0.0014
-2.1178
0.3406
0.1959
0.1826
-3.8701
0.5681
0.0157
0.0021
1105
0.0139
0.2777
-0.0001
-0.0008
0.3958
-0.0274
0.0642
-0.0620
-4.0701
0.1880
0.0091
0.0514
1106
-0.2513
0.4043
0.0031
-0.0164
0.4097
0.0060
0.1536
-0.0907
-5.0668
0.3261
0.0178
0.0419
1107
0.0670
0.3163
-0
-0.0145
0.2906
0.0485
0.2310
-0.0989
-5.0668
0.4016
0.0308
0.1474
1108
1.0112
0.6861
-0.0091
-0.0107
0.4281
0.0048
0.1337
-0.0933
-5.0668
0.4650
0.0304
0.1295
1109
0.5020
0.5370
-0.0030
-0.0375
0.3645
-0.0168
0.1078
-0.0641
-5.0668
0.4156
0.0311
0.1396
1110
0.1420
0.3295
0.0009
-0.0484
0.1728
-0.0189
-0.0164
-0.0230
-6.6650
0.4509
0.0099
0.0831Table10 shows the estimates of the ε suboptimal size m^k and the parameters m^k,β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, m^k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, m^k,β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k, for each of the energy commodity data sets. According to (58), the estimate γj,lm^k,k, j≠l, is positive if commodity pl is cooperating with commodity pj, and negative if commodity pl is competing with commodity pj. There is no interaction between the two commodities if γj,lm^k,k=0. It is apparent from the graph (from γ1,3m^k,k in Column 6) that coal is competing with natural gas during this period because the estimates of γ1,3m^k,k are mostly negative. It is apparent that natural gas and crude oil are either cooperating or competing, depending on the time period.The graph of the drift coefficient’s parameters with estimates in Table10 are given in Figure 11 below.Figure 11
The graph of interaction coefficientsγ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures11(a)–11(i) show the graph of the ε suboptimal interaction coefficient parameters γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k. According to (58), the estimate γj,lm^k,k, j≠l, is positive if commodity pl is cooperating with commodity pj, and negative if commodity pl is competing with commodity pj. There is no interaction between the two commodities if γj,lm^k,k=0. It is apparent from the graph of γ1,3m^k,k that coal is competing with natural gas because the estimates of γ1,3m^k,k are mostly negative. This shows that for fixed price p1 of natural gas, for every $1 increase in the price of coal, the price of natural gas decreases on the average by 0.0019. Also, it is apparent that natural gas and crude oil are either cooperating or competing, depending on the time period.Figures12(a)–12(c) are the graphs of β1m^k,k, β2m^k,k, and β3m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively.Figure 12
The graph ofβ1m^k,k, β2m^k,k, and β3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)Table11 gives the ε suboptimal estimates of the parameters u1m^k,k, σ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, u2m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, u3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k for each of the energy commodity data sets.Table 11
Estimatesσ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
σ1,1m^k,k
σ1,2m^k,k
σ1,3m^k,k
σ2,1m^k,k
σ2,2m^k,k
σ2,3m^k,k
σ3,1m^k,k
σ3,2m^k,k
σ3,3m^k,k
11
0
0
0
0
0
0
0
0.1303
0
12
0.0485
0.0004
0.0032
0.2734
0.0166
0
0.0513
0
0.0000
13
0.7333
0
0
0.9445
0
0
0.2489
0
0
14
0.2120
0.1386
0.0133
0.3877
0.4773
0.1195
0.1365
0.0665
0.0086
15
0.4246
0.1318
0.0021
0.03341
0.4894
0.1211
0.0112
0.6107
0.0696
16
0.5538
0.0778
0.1501
0.07751
0.2524
0.0811
0.0651
0.4251
0.0635
17
0.3907
0.0469
0.2230
0.08746
0.1848
0.2463
0
0.4458
0.0478
18
0.3523
0.0180
0.2178
0.04291
0.1877
0.1602
0.5681
0.0592
0.0115
19
0.5116
0.0619
0.2221
0.03266
0.2673
0.2465
0.4999
0.0569
0.0127
20
0.6431
0.0536
0.1866
0.0939
0.1700
0.0781
0.3789
0.3174
0.0046
…
…
…
…
…
…
…
…
…
…
495
0
0.0036
0.0406
0.2286
0.0600
0.0172
0.0110
0.9387
0.0182
496
0.1588
0.0035
0.0107
0.08183
0.3163
0.0102
0
0
0.0016
497
0.1551
0.0009
0.0065
0.07869
0.1453
0.4821
0.7777
0
0.0033
498
0.1576
0.0011
0.0073
0.0120
0.1679
0.3786
0.5334
0
0.0060
499
0.1197
0.0006
0.0059
0.0721
0.2391
0.0172
0.4405
0.1432
0.0097
500
0.3600
0.0001
0.0049
0.0273
0.3960
0.0079
0.6331
0.1410
0.0093
0.5010
0.0514
0.0033
0.0049
0.0182
0.3499
0.0111
0.7690
0.1376
0.0089
0.5020
0.2503
0.0034
0.0042
0.0222
0.1744
0.0132
0.6198
0.1274
0.0066
0.5030
0.1195
0.0147
0.0165
0
0.4283
0.0060
0
0.1530
0.0049
0.5040
0.0974
0.0144
0.0027
0
0.2241
0.0048
0.4778
0.0574
0.0043
0.5050
0.1422
0.0060
0.0131
0.0085
0.2023
0.0054
0.5604
0.0669
0.0004
…
…
…
…
…
…
…
…
…
…
1102
0.1898
0.0016
0.0413
0.8313
0.0767
0.0381
0.6875
0
0.1451
1103
0.2094
0.0015
0.0352
0.8262
0.0673
0.0451
0.7298
0.2808
0.0147
1104
0.1711
0.0011
0.0040
0.6648
0.0915
0.0462
0.5563
0.1831
0.0105
1105
0.1816
0.0012
0.0116
0.6658
0.1049
0.0371
0.6591
0.2874
0.0057
1106
0.1191
0.0011
0.0116
0.6260
0.1155
0.0393
0
0.0196
0.0060
1107
0.0417
0.0012
0.0041
0.4992
0.0781
0.0382
0.0271
0.2559
0.0065
1108
0.1058
0.0033
0.0045
0.0019
0.0589
0.0421
0.6209
0.8289
0.0018
1109
0.1740
0.0021
0
0.0305
0.0446
0.0316
0.8431
0.2366
0.4511
1110
0.2912
0.0021
0.0163
0.0385
0.0342
0.0037
0.2910
0.0489
0.0257Figures13(a)–13(c) are the graphs of σ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively.Figure 13
The graph ofσ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Table12 shows the real and simulated estimates for the spot price processes pjt, j∈I1,n.Table 12
Real and simulated estimates (with jump) for natural gas, crude oil, and coal.
tk
Natural gas
Crude oil
Coal
Real
Simulatedp1sm^k,k
Real
Simulatedp2sm^k,k
Real
Simulatedp3sm^k,k
11
4.0200
4.0500
58.9900
58.5200
16.5900
16.6000
12
3.9900
3.9600
59.5200
59.5099
17.4600
17.4635
13
3.7500
3.6690
61.4500
60.3377
17.8900
17.8886
14
3.7700
3.7341
60.4900
59.4191
17.5500
17.5188
15
3.4100
3.3967
61.1500
60.1580
17.4100
17.4188
16
3.3500
3.3947
62.4800
62.5028
16.7500
16.7789
17
3.4900
3.3900
63.4100
63.5524
17.6600
17.5341
18
3.5500
3.4957
65.0900
64.9224
17.5200
17.8130
19
3.9200
3.8743
66.3100
66.8239
18.5000
18.8453
20
3.8600
3.8241
68.5900
68.1206
19.0600
18.9453
…
…
…
…
…
…
…
494
4.2300
4.2295
106.7000
106.6374
33.2200
33.1852
495
4.1900
4.2191
107.1800
106.9973
32.7600
32.6677
496
4.3300
4.3414
110.8400
111.0084
33.6500
33.8070
497
4.3300
4.3261
111.7200
111.7084
33.7100
33.7061
498
4.3700
4.3863
111.6800
111.2084
34.7500
35.7057
499
4.3200
4.2167
111.7200
111.6126
34.5400
33.5457
500
4.3500
4.3577
112.3100
112.4358
34.0400
34.2862
501
4.3800
4.3535
112.3800
112.5682
33.1000
33.1330
502
4.5100
4.4389
113.3900
113.2925
33.6700
33.6216
503
4.6000
4.6139
113.0300
113.3077
33.9400
33.9216
504
4.6000
4.5964
110.6000
110.8350
33.8300
33.9216
505
4.5900
4.5564
108.7900
108.7598
32.0200
31.9867
…
…
…
…
…
…
…
1102
3.7200
3.6616
108.2300
108.2354
4.7700
4.5482
1103
3.7300
3.6477
106.2600
106.1451
5.0100
5.1120
1104
3.6800
3.6748
104.7000
104.6723
4.9800
5.1180
1105
3.6600
3.6861
103.6200
104.6723
4.7300
4.7893
1106
3.5900
3.6436
103.2200
102.9765
4.6800
4.7074
1107
3.5200
3.5213
102.6800
102.7652
4.6300
4.6419
1108
3.4900
3.4564
103.1000
102.9765
4.7400
4.4016
1109
3.5100
3.2596
102.8600
102.9652
4.3300
4.1826
1110
3.4800
3.4604
102.3600
102.4345
4.1800
4.4606Figure14 shows the graph of the real and simulated prices (with jump) for natural gas, crude oil, and coal data set.Figure 14
Real and simulated prices (with jump) for natural gas, crude oil, and coal.
(a)
(b)
(c)Figures14(a)–14(c) show the graph of the real and simulated spot prices for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively. The red line represents the real data set p, while the blue line represent the simulated data set pm^k,ks. It is clear that the graph fits well. To reduce magnitude of error, we increase the magnitude of time delay. It is obvious that these curves fit better than the curves in Figure 7. It follows that the interconnected dynamical system with jump process incorporated into it performs better than the one without jump.


## 7.1. Coordination of Data Observation, Iterative Process, and Simulation Schedules

Without loss of generality, we assume that the real data observation/collection partition schedulesℙi−1, i∈I1,K∗ are defined in (41). We present definitions of iterative process and simulation time schedule following the definitions in Otunuga et al. [16].Definition 14.
The iterative process time schedule in relation with the real data collection schedule is defined by(74)Iℙi−1=F−ri−1tki−1:fortki−1∈ℙi−1,fori∈I1,K∗,k∈I−ri−1,Ni−1,where F−ri−1tki−1=tk−ri−1i−1 is a forward shift operator [32].
The simulation time is based on the orderdi−1 of the time series model of mki−1 local conditional sample mean and covariance processes in (48).Remark 15.
For the case whereK=0, we have Iℙi−1=Iℙ, where ℙi−1=ℙ is defined in (40). This is the iterative time schedule in the absence of jumps.Definition 16.
The simulation process time schedule in relation with the real data observation schedule is defined by(75)Sℙi−1=Fri−1tki−1:fortki−1∈ℙi−1,ifdi−1≤ri−1,Fdi−1tki−1:fortki−1∈ℙi−1,ifdi−1>ri−1,k∈I−ri−1,Ni−1.Remark 17.
For eachi∈I1,K∗, the initial times of iterative and simulation processes are equal to the real data times tri−1i−1 and tdi−1i−1, whenever di−1≤ri−1 and di−1>ri−1, respectively. The iterative process and simulation process times with jump are tk+ri−1i−1 and tk+di−1i−1, i∈I1,K∗, respectively.


## 7.2. Conceptual Computational Parameter Estimation Scheme

For the conceptual computational dynamic system parameter estimation, we need to introduce a few concepts of local admissible sample/data observation sizemki−1 local admissible conditional finite sequence at tki−1∈Sℙi−1 and local finite sequence of parameter estimates at tki−1.Definition 18.
For eachi∈I1,K∗, and tki−1∈ITi−1−τi−1,Ti, we define local admissible sample/data observation size mki−1 at tki−1 as mki−1∈OSki−1, where
(76)OSki−1=I2,ri−1+Si−1+k−1,ifdi−1≤ri−1,I2,di−1+Si−1+k−1,ifdi−1>ri−1,k∈I0,Ni−1.
Moreover,OSki−1 is referred as the local admissible set of lagged sample/data observation size at tki−1.Remark 19.
We note that ifK=0, Si−1=0, the point tki−1=tk∈t0,T. Thus, (76) is reduced to
(77)OSki−1=OSk=I2,r+k−1,ifd≤r,I2,d+k−1,ifd>r,k∈I0,N.Definition 20.
For eachi∈I1,K∗, mki−1∈OSki−1 in Definition 18 and k∈I0,Ni−1, a mki−1 local admissible lagged adapted finite restriction sequence of conditional sample/data observation at time tki−1 to subpartition Pki−1 of ℙi−1 in Definition 8 is defined by Eyi−1tli−1Fl−1i−1l=k−mki−1k−1,Epi−1tli−1Fl−1i−1l=k−mki−1k−1. Moreover, a mki−1 class of admissible lagged adapted finite sequences of conditional sample/data observation of size mki−1 at tki−1 is defined by
(78)ASki−1=Eyi−1tli−1Fl−1i−1l=k−mki−1k−1mki−1∈OSki−1,Epi−1tli−1Fl−1i−1l=k−mki−1k−1mki−1∈OSki−1.In the case of energy commodity model, for eachi∈I1,K∗, mki−1∈OSki−1, we find corresponding mki−1 local admissible adapted finite sequence of conditional sample/data observation at tki−1, Eyi−1tli−1Fl−1i−1l=k−mki−1k−1,Epi−1tli−1Fl−1i−1l=k−mki−1k−1. For i∈I1,K∗, using this sequence and solutions of (70), we compute
(79)uji−1mk,tki−1,βji−1mki−1,tki−1,κj,li−1mki−1,tki−1,γj,li−1mki−1,tki−1,δj,li−1mki−1,tki−1,σj,li−1mki−1,tki−1,k∈0,Ni−1,for j,l∈I1,n. This leads to a local finite sequence of parameter estimates at tki−1 defined on OSki−1 as follows:
(80)u^ji−1mki−1,tki−1,β^ji−1mki−1,tki−1,κ^j,li−1mki−1,tki−1,γ^j,li−1mki−1,tki−1,δ^j,li−1mki−1,tki−1,σ^j,li−1mki−1,tki−1mki−1∈OSki−1.The above defined collection is denoted by(81)Uk,Bk,Kk,γk,δk,σk=u^ji−1mki−1,tki−1,β^ji−1mki−1,tki−1,κ^j,li−1mki−1,tki−1,γ^j,li−1mki−1,tki−1,δ^j,li−1mki−1,tki−1,σ^j,li−1mki−1,tki−1mki−1∈OSki−1,forj∈I1,n,i∈I1,K∗.


## 7.3. Conceptual Computation of State Simulation Scheme

For the development of a conceptual computational scheme, we need to employ the method of induction. The presented simulation scheme is based on the idea of lagged adaptive expectation process [17]. For j,l∈I1,n, an autocorrelation function (ACF) analysis [21, 32] performed on smki−1,kj,jy,smki−1,kj,jp suggests that the interconnected discrete-time dynamic model of local conditional sample mean and sample variance statistics in (48) is of order di−1=2. In view of this, we need to identify the initial data. We begin with a given initial data yi−1Ti−1,pi−1Ti−1, ∑m0i−1,t0i−1ym0i−1∈OS0i−1,∑m0i−1,t0i−1pm0i−1∈OS0i−1,∑m−1i−1,t−1i−1ym−1i−1∈OS−1i−1,∑m−1i−1,t−1i−1pm−1i−1∈OS−1i−1, S¯m−1i−1,t−1i−1ym−1i−1∈OS−1i−1,S¯m−1i−1,t−1i−1pm−1i−1∈OS−1i−1 as defined in (45), (46), (47), and (48).Letysmki−1,tki−1,psmki−1,tki−1≡yi−1,smki−1,tki−1,pi−1,smki−1,tki−1 be a simulated value of Eyi−1tki−1Fk−1i−1,Epi−1tki−1Fk−1i−1 at time tki−1 corresponding to an admissible sequence Eyi−1tli−1Fl−1i−1,Epi−1tli−1Fl−1i−1l=k−mkk−1∈ASki−1. For q=1, and j∈I1,n, the simulated value yjsmki−1,tki−1,pjsmki−1,tki−1 is generated from the discretized Euler scheme (62) as follows:
(82)yjsmki−1,tki−1=yjsmk−1i−1,tk−1i−1+uji−1mk−1i−1,tk−1i−1−yjsmk−1i−1,tk−1i−1κj,ji−1mk−1i−1,tk−1i−1yjsmk−1i−1,tk−1i−1+∑l≠jnκj,li−1mk−1i−1,tk−1i−1ylsmk−1i−1,tk−1i−1Δt+uji−1mk−1i−1,tk−1i−1−yjsmk−1i−1,tk−1i−1δj,ji−1mk−1i−1,tk−1i−1ΔWj,jmki−1,tki−1+∑l≠jnδj,li−1mk−1i−1,tk−1i−1ylsmk−1i−1,tk−1i−1ΔWj,lmki−1,tki−1,tki−1∈Ti−1,Ti,pjsmki−1,tki−1=pjsγj,ji−1mk−1i−1,tk−1i−1yjsmk−1i−1,tk−1i−1−pjsmk−1i−1,tk−1i−1+βji−1mk−1i−1,tk−1i−1+∑l≠jnγj,li−1mk−1i−1,tk−1i−1plsmk−1i−1,tk−1i−1Δt+σj,ji−1mk−1i−1,tk−1i−1pjsmk−1i−1,tk−1i−1ΔZj,jmki−1,tki−1+pjsmk−1i−1,tk−1i−1∑l≠jnσj,li−1mk−1i−1,tk−1i−1plsmk−1i−1,tk−1i−1ΔZj,lmki−1,tki−1,tki−1∈Ti−1,Ti.To find the simulated valueyji,sTi and pji,sTi, we need to estimate π^ji and θ^ji by first simulatinglimt→Ti−yji−1t,Ti−1,yi−1,s≡yjsmNi−1i−1,tNi−1i−1 and limt→Ti−pji−1t,Ti−1,yi−1,s,pi−1,s≡pjsmNi−1i−1,tNi−1i−1 as follows:
(83)yjsmNi−1i−1,tNi−1i−1=yjsmNi−1−1i−1,tNi−1−1i−1+uji−1mNi−1−1i−1,tNi−1−1i−1−yjsmNi−1−1i−1,tNi−1−1i−1∑l=1nκj,li−1mNi−1−1i−1,tNi−1−1i−1ylsmNi−1−1i−1,tNi−1−1i−1Δt+uji−1mNi−1−1i−1,tNi−1−1i−1−yjsmNi−1−1i−1,tNi−1−1i−1δj,ji−1mNi−1−1i−1,tNi−1−1i−1ΔWj,jmNi−1i−1,tNi−1i−1+∑l≠jnδj,li−1mNi−1−1i−1,tNi−1−1i−1ylsmNi−1−1i−1,tNi−1−1i−1ΔWj,lmNi−1i−1,tNi−1i−1,pjsmNi−1i−1,tNi−1i−1=pjsγj,ji−1mNi−1−1i−1,tNi−1−1i−1yjsmNi−1−1i−1,tNi−1−1i−1−pjsmNi−1−1i−1,tNi−1−1i−1+βji−1mNi−1−1i−1,tNi−1−1i−1+∑l≠jnγj,li−1mNi−1−1i−1,tNi−1−1i−1plsmNi−1−1i−1,tNi−1−1i−1Δt+σj,ji−1mNi−1−1i−1,tNi−1−1i−1pjsmNi−1−1i−1,tNi−1−1i−1ΔZj,jmNi−1i−1,tNi−1i−1+pjsmNi−1−1i−1,tNi−1−1i−1∑l≠jnσj,li−1mNi−1−1i−1,tNi−1−1i−1plsmNi−1−1i−1,tNi−1−1i−1ΔZj,lmNi−1i−1,tNi−1i−1.From this, we calculateπ^ji and θ^ji as:
(84)π^ji=Eyji−1TiFTi−1i−1yjsmNi−1i−1,tNi−1i−1θ^ji=Epji−1TiFTi−1i−1pjsmNi−1i−1,tNi−1i−1.Thus,yji,sTi=π^jiyji−1,sTi−,Ti−1,yi−1,s and pji,sTi=θ^jipji−1,sTi−,Ti−1,yi−1,s,pi−1,s. Let (ysmki−1,tki−1mki−1∈OSki−1, psmki−1,tki−1mki−1∈OSki−1) be a mki−1- local sequence of simulated values corresponding to mki−1-admissible lagged adapted finite sequence of conditional observation belonging to ASki−1, and corresponding term of sequence.(Uk,Bk,Kk,γk,δk,σk). Thus, for each i∈I1,K∗, (ysmki−1,tki−1mki−1∈OSki−1, psmki−1,tki−1mki−1∈OSki−1) are the finite sequence correspondence of simulated values of Eyi−1tki−1Fk−1i−1,Epi−1tki−1Fk−1i−1 at tki−1.


## 7.4. Mean-Square Suboptimal Procedure

To find the best estimate ofEytki−1Fk−1i−1,Eptki−1Fk−1i−1 using a local admissible finite sequence ysmki−1,tki−1mki−1∈OSki−1,psmki−1,tki−1mki−1∈OSki−1, we need to compute a finite sequence of quadratic mean square error corresponding to ysmki−1,tki−1mki−1∈OSki−1,psmki−1,tki−1mki−1∈OSki−1. The quadratic mean square error is defined below.Definition 21.
For eachi∈I1,K∗, the quadratic mean square error of Eytki−1Fk−1i−1,Eptki−1Fk−1i−1 relative to each member of the term of local admissible sequence ysmki−1,tki−1mki−1∈OSki−1,psmki−1,tki−1mki−1∈OSki−1 of simulated values is defined by
(85)Ξmki−1,tki−1=ysmki−1,tki−1−Eyi−1tki−1Fk−1i−12+psmki−1,tki−1−Epi−1tki−1Fk−1i−12.For any arbitrary small positive numberε and for each time tki−1, to find the the best estimate from the admissible simulated values of simulated sequence of ysmki−1,tki−1mki−1∈OSki−1,psmki−1,tki−1mki−1∈OSki−1 for Eytki−1Fk−1i−1,Eptki−1Fk−1i−1, we determine the following sub-optimal admissible set of mki−1-size local conditional sample
(86)Mtki−1=mki−1∈OSki−1:Ξmki−1,tki−1<ε,fori∈I1,K∗.Among these collected values, the value that gives the minimumΞmki−1,tki−1 for k∈0,Ni−1 are recorded as m^ki−1. If more than one value exist, then the largest of such mki−1’s is recorded as m^ki−1. If condition (86) is not met at time tki−1, the value of mki−1 where the minimum minmki−1Ξmki−1,tki−1 is attained is recorded as m^ki−1. The ε− level sub-optimal estimates of the parameters u^ji−1mki−1,tki−1,κ^j,li−1mki−1,tki−1,β^ji−1mki−1,tki−1,δ^j,li−1mki−1,tki−1,γ^j,li−1mki−1,tki−1,σ^j,li−1mki−1,tki−1 are recorded as uji−1m^k,yji−1,k,κj,li−1m^k,yji−1,k,βji−1m^k,pji−1,k,δj,li−1m^k,yji−1,k,γj,li−1m^k,pji−1,k,σj,li−1m^k,pji−1,k. Finally, the simulated value ysmki−1,tki−1, psmki−1,tki−1 at time tki−1 with m^ki−1 is now recorded as the best estimate for Eyi−1tki−1∣Fk−1i−1 and Epi−1tki−1∣Fk−1i−1. The value ysm^ki−1,k, psm^ki−1,k is called the ε− sub-optimal simulated value of ysmki−1,tki−1 and psmki−1,tki−1 of Eyi−1tki−1∣Fk−1i−1 and Epi−1tki−1∣Fk−1i−1 at tki−1.


## 7.5. Illustration: Application of Conceptual Computational Algorithm to Energy Commodity Data Set

In this subsection, we apply the above conceptual computational algorithm to study the relationship between three energy commodities by settingn=3 in (56). The three energy commodities are daily Henry Hub Natural gas data set, daily crude oil data set, and daily coal data set for the period of 05/04/2009-01/03/2014, [35–37]. Thus, for each pair y1,p1, y2,p2, and y3,p3, the drift and diffusion coefficient function of the stochastic dynamic equation governing yj,pj, for j∈I1,3 has 4 and 3 parameters each to be estimated, respectively. Thus, there are 42 parameters to be estimated in total. Using Δt=1, ε=0.001, for each j∈I1,3, the ε− level sub-optimal estimates of parameters uji−1m^ki−1,k, βji−1m^ki−1,k, κj,li−1m^ki−1,k, γj,li−1m^ki−1,k, δj,li−1m^ki−1,k, σj,li−1m^ki−1,k, l∈I1,3, at each real data times are exhibited below.


### 7.5.1. Illustration: Relationship between Natural Gas, Crude Oil, and Coal: Without Incorporating Jump Process

In this subsubsection, we analyze the relationship between natural gas, crude oil, and coal without the jump process. Forj,l∈I1,3, the stochastic dynamic system governing the three energy commodities is described in (59) of Remark 12. Here, (y1,p1) denotes the mean spot and the spot price process of natural gas, (y2,p2) denotes the mean spot and the spot price process of crude oil, and (y3,p3) denotes the mean spot and the spot price process of coal.Using the discretized scheme (82), we apply the above conceptual computational algorithm for the real-time data sets namely daily Henry Hub natural gas data set, daily crude oil data set, and daily coal data set for the period of 05/04/2009-01/03/2014, [35–37]. Using Δt=1, ε=0.001, r=10, and d=2, the ε-level suboptimal estimates of the parameters at each real data times are described below for each commodity data sets.The parameters corresponding to the natural gas data set areu1m^k,k, β1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, σ1,1m^k,k, σ1,2m^k,k, and σ1,3m^k,k. The parameters corresponding to the crude oil data set are u2m^k,k, β2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, and σ2,3m^k,k. The parameters corresponding to coal data set are u3m^k,k, β3m^k,p3,k, κ3,1m^k,k, κ3,2m^k,k, κ3,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, γ3,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, δ3,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k.The following table gives the drift coefficient’s parameter estimatesu1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for the decoupled dynamical system for y in the case where jump is not incorporated into the dynamical system.Table1 shows the estimates of the ε suboptimal size m^k, j∈I1,3, the parameters u1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for each of the energy commodity data sets. Moreover, d≤r and the initial real data time is tr=t10.Table 1
Estimatesm^k, u1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k.
tk
m^k
Natural gas
Crude oil
Coal
u1m^k,k
κ1,1m^k,k
κ1,2m^k,k
κ1,3m^k,k
u2m^k,k
κ2,1m^k,k
κ2,2m^k,k
κ2,3m^k,k
u3m^k,k
κ3,1m^k,k
κ3,2m^k,k
κ3,3m^k,k
×10−16
×10−18
×10−18
×10−18
×10−18
×10−18
11
1
4.1593
0.0211
0
0
57.7000
0
0
0
16.7407
0
0
0
12
3
4.2000
0.0111
0
0
58.6313
0.0011
0.0310
-0.0012
16.2395
0
0
-0.0376
13
5
4.0616
0.0679
-0.0054
-0.0035
58.5378
-0.0035
0.0205
0.0032
16.2680
0
0
0.1069
14
5
4.0616
-0.0242
-0.0179
0
61.4809
0.0020
0.0098
0
15.5249
0
0
-0.0294
15
8
4.0910
0.6416
-0.2898
0
58.9282
-0.0036
0.0128
0.0071
16.8286
0
0
0.0513
16
8
4.0160
0.2101
0
0
59.6867
-0.0051
0.0080
0.0071
17.0888
0
0
0.0415
17
8
4.9575
0.1876
0
0
60.6244
0.0024
0.0052
0
17.4120
-0.0003
0.0001
0.0555
18
8
4.9575
-0.1947
0
0
61.0700
0
0
0
17.2374
0
-0
0
19
6
4.7336
-1.4476
5.8820
0
61.9414
0
0.0043
-0.0086
16.8438
0.0001
0.0001
0.0768
20
6
2.5646
0.3319
0.7261
0
62.7899
0
0.0053
0.0082
18.3022
-0.0083
0.0027
0.0558
…
…
…
…
…
…
…
…
…
…
…
…
…
…
495
8
3.9654
0.0591
-0
0
108.2457
0.0038
0.0049
-0.0023
33.1313
0.0027
0.0009
0.0363
496
5
4.0421
0.0616
0.0001
0.0017
107.5186
0
0
0
33.4224
-0.0005
0.0003
0.0214
497
6
4.0514
0.0127
-0.0002
0.0020
109.8836
0
0
-0.0001
33.3388
0.0002
0
0.0443
498
7
4.1646
0.0442
-0.0012
-0.0053
107.8013
-0.0021
0.0033
0.0038
33.2862
-0.0002
0.0006
0.0343
499
6
4.1226
0.0352
-0.0020
0
108.1554
-0.0005
0.0032
0.0039
33.2862
0.0010
0.0001
0.0068
500
6
4.2625
0.0733
-0.0002
0
110.5101
-0.0032
0.0033
0.0016
36.1647
0.0003
0.0003
0.0079
501
8
3.1551
0
0
-0.0009
110.3071
0.0014
0.0025
0
34.7467
0
0
0
502
4
4.1564
0.0914
-0.0002
0
111.1186
0
0.0013
-0.0031
49.4050
0.0026
-0.0002
0.0211
503
5
4.5799
0.0467
0.0004
0
112.0057
0
0.0027
-0.0043
34.7207
-0.0001
-0.0001
0.0216
504
4
4.3061
0.0236
0.0002
0.0007
112.3186
0
0.0021
0.0015
34.4483
0.0019
0.0003
0.0170
505
9
4.4325
-0.0015
-0.0018
0.0030
106.3345
0
0.0043
0.0001
33.7160
0
-0.0006
0.0265
…
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
7
3.5429
-0.0286
-0.0006
-0.0028
110.3777
0.0006
0.0045
0
5.2399
0
0.0013
0.0008
1103
4
3.5601
0.1028
0.0001
0.0001
111.1585
-0.0003
0.0083
0
5.4824
0
0.0077
0.0485
1104
4
3.5314
0.0809
0.0018
0.0090
109.0996
-0.0007
0.0095
0.0013
11.0949
-0.0018
0.0005
0.1175
1105
4
3.4439
0.1551
-0.0008
-0.0015
106.5667
0.0033
0.0073
-0.0020
4.8300
-0.0012
-0.0003
0.1283
1106
6
3.8206
0.2258
0.0004
0
104.7497
0
0
0.0027
4.8300
0
0.0008
0
1107
4
3.6917
0.2132
-0.0001
-0.0008
105.1229
0.0011
0.0039
0
4.3586
-0.0005
0.0004
0.1418
1108
5
3.7871
0
0
0
105.3595
0.0006
0.0027
-0.0009
4.8000
0.0006
-0.0001
0.1265
1109
4
3.8445
-0.0405
-0.0011
0.0011
102.9022
-0.0044
0.0037
0.0039
5.0279
0
0
0
1110
5
3.8399
0.0212
0.0004
0
102.8313
-0.0020
0.0045
0.0018
4.6817
0.0021
0.0041
0.0536Figure1 shows the drift coefficient’s parameter estimates u1m^k,k, u2m^k,k, and u3m^k,k for the decoupled dynamical system for y in the case where jump is not incorporated into the dynamical system.Figure 1
The graph of mean levelu1m^k,k, u2m^k,k, and u3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)Figures1(a)–1(c) are the graphs of u1m^k,k, u2m^k,k, and u3m^k,k against time tk for the daily Henry Hub natural gas price [36], daily crude oil price [37], and daily coal price [35] data set, respectively. By plotting the real data sets, it is easily seen that the graphs of u1m^k,k, u2m^k,k, and u3m^k,k are similar to the graph of the Henry Hub natural gas, crude oil, and coal data set, respectively. We expect this to happen because uj, j∈I1,3, are the equilibrium spot price processes of (5). The sample means of the real spot price data sets for natural gas, crude oil, and coal are given by 3.7218, 88.5620, and 16.5543, respectively. It can be seen from Figures 1(a)–1(c) that the graphs of u1m^k,k, u2m^k,k, and u3m^k,k move around the mean values 3.7218, 88.5620, and 16.5543, respectively. This analysis shows that the parameters uj, j∈I1,3, are statistic processes for the respective mean of the data sets at time tk.The graph of the interaction parametersκ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for the decoupled dynamical system for y (with no jump incorporated into the dynamical system) are given in Figure 2 below.Figure 2
The graph of interaction coefficientsκ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures2(a)–2(i) show the graph of the ε suboptimal interaction coefficient parameters κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k. The interaction coefficients κj,l, j≠l are negligible, because each estimate is <<10−15. Thus, this shows that the model describing the mean spot price, yj, is mainly characterized by the market potential κj,juj−yjyj, j∈I1,n.Table2 below shows the estimates of the diffusion coefficient’s parameters δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k,δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k for the decoupled dynamical system for y in the case where jump is not incorporated into the dynamical system.Table 2
Estimatesδ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k,δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
δ1,1m^k,k
δ1,2m^k,k
δ1,3m^k,k
δ2,1m^k,k
δ2,2m^k,k
δ2,3m^k,k
δ3,1m^k,k
δ3,2m^k,k
δ3,3m^k,k
11
0.0123
0.0012
0.0001
0
0.0223
0
0.0412
0
0.0022
12
0.0024
0.0011
0.0121
0.0234
0.0245
0
0
0
0.0112
13
0.0001
1.3425
1.7280
1.9811
0.9899
0.9731
0.6374
0.6374
0.0123
14
0
1.1267
0.6027
2.3258
0.1213
3.9128
1.6564
1.6564
0.0004
15
1.15260
0.4287
0.6210
2.3252
0.0006
0.5083
1.6650
1.6650
0.4565
16
4.9354
0
0
2.3217
0.0120
1.1124
1.6724
1.6724
0.8762
17
4.1360
0.0989
3.6877
1.6425
0
0
1.7719
1.7719
0
18
3.0410
0.1527
0
1.3105
0.9167
1.3451
1.7630
1.7630
0
19
2.7713
0
0
1.1052
0
3.3241
1.7400
1.7400
0
20
2.8461
0.0012
0.2221
0.1196
5.1929
0
0.6532
0.9876
0.0082
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
494
2.9961
0.0586
0
0.5529
0
0.42339
0
0
0.5187
495
5.9059
0
0.0584
0.5488
0.8947
0
0.0017
0.0021
0.0001
496
0.1121
0
0.6613
0.5767
0.9899
0
0.8763
0
0.9827
497
1.1229
0.0095
0.0988
0.6499
5.8547
0
1.1317
1.1317
0.0012
498
0.6946
0.0101
0
0
5.8298
0.0320
1.0294
1.0294
0.0321
499
0.7353
0.0066
0.0384
0
5.7180
0.0330
0.7317
0.7317
0.0431
500
1.7509
0.0069
0.0283
0.4307
5.6133
0.0413
0.4826
0.4826
0.0783
501
2.1299
0.0077
0.0282
0.5043
5.6282
0.0308
0.4272
0.4272
0.0002
502
0.9778
0.0077
0.0255
0.2878
4.6543
0.0322
0.5239
0.5239
0.0098
503
0.9872
0
0
0.2909
4.5544
0.0411
1.4523
1.4523
0.0087
504
1.1329
0
0
0.3707
0
0.1128
2.4181
2.4181
0
505
1.9178
0
0
0.3812
1.3243
0.1724
4.9207
4.9207
0
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
0
0.0331
0.0056
0.9297
3.9502
0
0.2853
1.8033
1.1355
1103
1.5077
0.0626
0.0332
1.1017
2.8221
0
0
0
1.4133
1104
0
0.0435
0.5821
0.1939
4.5585
0
0
0
1.1672
1105
0
0
1.52970
0.1922
3.2418
0.7273
0.2726
0.2726
0
1106
4.4476
0.323
0.5112
3.5487
3.8113
1.0179
0.3296
0.3296
0
1107
2.4312
0.0011
0.0435
0.2001
2.6026
0.9354
0
0
1.7245
1108
2.5079
0.1232
0.4542
0.3781
0
0.8825
0.1878
0.1878
0
1109
1.7828
0.0431
0.3210
0.4024
0
0.8812
0
0
1.3191
1110
1.2706
0.0056
1.1123
0.3252
0
0.8078
0
0
1.0233The graph of the diffusion coefficient’s parameter for the decoupled dynamical system fory without jump incorporated into the dynamical system are given in Figure 3 below.Figure 3
The graph of interaction coefficientsδ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures3(a)–3(i) show the graphs of the ε suboptimal interaction measure of fluctuation coefficient parameters δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.Table3 gives the drift coefficient’s parameter estimates β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k for the dynamical system for p without jump.Table 3
Estimatesm^k, β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, m^k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, m^k,β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
β1m^k,k
γ1,1m^k,k
γ1,2m^k,k
γ1,3m^k,k
β2m^k,k
γ2,1m^k,k
γ2,2m^k,k
γ2,3m^k,k
β3m^k,k
γ3,1m^k,k
γ3,2m^k,k
γ3,3m^k,k
11
0.3462
0.2579
-0.0039
0.0218
0.0029
0.0023
0.0056
0.0124
0
0
0.0432
0.0012
12
0.1681
0.3497
-0.0109
0.0248
0
0
0
0
0.6812
-0.3513
0.0248
0
13
0.1592
0.3755
-0.0102
0.0228
-0.0490
0.0228
0
-0.0027
1.0795
-0.2904
0.0135
0
14
-0.0889
0.5001
-0.0069
0.0257
-0.3700
-0.1689
0
0.0633
-0.6689
-0.1969
0.0166
0.0177
15
0.7025
0.6513
-0.0242
0.0376
-0.0903
-0.1737
0
0.0477
-0.8220
-0.0944
0.0061
0.0291
16
0.6727
0.6513
0.0188
-0.1048
0.3425
-0.1847
0
0.0252
-0.0480
0.0694
-0.0112
0.0165
17
0.3253
0.3674
-0.0103
0.0140
-0.4531
-0.1457
0
0.0638
-3.6240
0.1086
0.0266
0.0569
18
0.1523
0.3433
0.0014
-0.0163
4.0859
0.2320
0.3114
-0.2889
-1.3277
-0.0566
0.0088
0.0353
19
-7.9573
0.3433
-0.2058
1.1496
0.1389
-0.0677
0.0004
0.0065
-0.9232
0.0368
-0.0277
0.0883
20
0.0514
0.3028
0.0041
-0.0192
-0.1464
-0.0409
0.0004
0.0178
-1.9138
0.0530
-0.0104
0.0836
…
…
…
…
…
…
…
…
…
…
…
…
…
495
0.0552
0.2994
0.0015
-0.0063
1.8435
-0.1608
0.1040
-0.0368
0.5815
-0.3597
-0.0050
0.0350
496
0.1799
0.1860
0.0004
-0.0067
1.5682
-0.0268
0.4723
-0.0502
0.8586
-0.2029
-0.0094
0.0241
497
0.8047
0.1923
0.0023
-0.0314
6.6699
-0.2655
0.4723
-0.1649
0.7191
-0.2061
-0.0053
0.0172
498
0.2742
0.2651
0.0020
-0.0145
1.0042
0.0088
0.0226
-0.0315
0.3978
-0.1680
-0.0034
0.0161
499
0.4915
0.2295
-0.0006
-0.0125
1.3074
0.3761
0.0073
-0.0872
0.1425
-0.1899
-0.0026
0.0224
500
0.5659
0.1618
0.0008
-0.0194
0.4040
-0.0889
0.0392
-0.0011
0.3674
-0.2313
-0.0073
0.0331
501
0.4498
0.1679
0.0010
-0.0167
0.4230
-0.1297
0.0434
0.0035
-0.9002
1.1703
-0.0775
0.1011
502
0.4836
0.1850
-0.0001
-0.0139
0.5570
-0.1502
0.0384
0.0022
-0.2313
0.6524
-0.0496
0.0663
503
0.4696
0.1850
0.1224
-0.0919
-0.0441
0.0299
0.0384
-0.0023
3.7804
0.0120
-0.0498
0.0389
504
-0.0456
0.1850
0.0088
-0.0270
0.6112
-0.0820
0.0425
-0.0080
6.4696
0.4005
-0.0950
0.0543
505
0.0464
1.7125
-0.0423
0.1339
0.7135
-0.1115
0.1135
-0.0082
2.2295
0.0897
-0.0357
0.0306
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
0.6765
0.0455
-0.0020
-0.0908
0.2863
-0.2183
0.1891
0.1028
4.3927
3.8144
0.1072
0.0250
1103
1.1804
0.4214
-0.0149
0.0837
-2.1858
0.4491
0.1891
0.1135
-6.1960
0.7446
0.0261
0.0144
1104
0.1069
0.2489
-0.0009
-0.0014
-2.1178
0.3406
0.1959
0.1826
-6.4415
0.0339
-0.0037
0.1429
1105
0.2367
0.0128
0.0065
-0.0019
0.2633
0.0565
0.0742
-0.0997
0.6510
2.4930
0.0714
0.1429
1106
0.1178
0
0
-0.0014
0.1384
0.0784
0.2014
-0.0904
0.6510
2.4930
0.6341
0.1429
1107
0.1466
0.4648
-0.0002
-0.0271
0.5787
-0.0297
0.1305
-0.0979
-4.5897
-0.0971
0.0240
0.0509
1108
0.3240
0.2478
0.1212
-0.0074
0.4293
-0.0678
0.0721
-0.0389
-4.5961
-0.0734
0.0233
0.0507
1109
0.121
0
0
-0.0021
0.2282
-0.0468
0.0721
-0.0133
-4.5961
-2.6776
0.5625
0.4702
1110
0.002
0
0
-0.0056
0.1523
0.0121
0.0011
-0.0129
9.5959
0.9045
-0.1478
0.0499Table3 shows the estimates of the ε suboptimal size m^k, j∈I1,3 and the parameters β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k, for each of the energy commodity data sets. Moreover, d≤r, and the initial real data time is tr=t10.In the following, we give the graph of the drift coefficient’s parameters with estimates in Table3 in Figure 4 below.Figure 4
The graph of interaction coefficientsγ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures4(a)–4(i) show the graphs of the ε suboptimal interaction coefficient parameters γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k. According to (58), the estimate γj,lm^k,k, j≠l, is positive if commodity pl is cooperating with commodity pj, and negative if commodity pl is competing with commodity pj. There is no interaction between the two commodities if γj,lm^k,k=0. It is apparent from the graph of γ1,3m^k,k that coal is competing with natural gas because the estimates of γ1,3m^k,k are mostly negative. Also, it is apparent that natural gas and crude oil are either cooperating or competing, depending on the time period.Figures5(a)–5(c) are the graphs of β1m^k,k, β2m^k,k, and β3m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively.Figure 5
The graph ofβ1m^k,k, β2m^k,k, and β3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)Table4 gives the ε suboptimal estimates of the parameters σ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k for each of the energy commodity data sets.Table 4
Estimatesσ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
σ1,1m^k,k
σ1,2m^k,k
σ1,3m^k,k
σ2,1m^k,k
σ2,2m^k,k
σ2,3m^k,k
σ3,1m^k,k
σ3,2m^k,k
σ3,3m^k,k
11
0
0
0
0
0
0
0
0
0
12
0.0485
0.0004
0.0032
0.2734
0.0166
0
0
0
0.0000
13
0
0
0
0
0
0
0
0
0
14
0.2120
0.1386
0.0133
1.2573
0.4773
0.1195
0
0.0665
0.0086
15
0.4246
0.1318
0.0021
2.1081
0.4894
0.1211
0
0.6107
0.0696
16
0.5538
0.0778
0.1501
0
0.2524
0.0811
0.0651
0.4251
0.0635
17
1.1121
0.0469
0.2230
0
0.1848
0.2463
0
0.4458
0.0478
18
1.5347
0.0180
0.2178
0
0.1877
0.1602
0.5681
0.0592
0.0115
19
1.1315
0.0619
0.2221
0
0.2673
0.2465
0.4999
0.0569
0.0127
20
2.0845
0.0536
0.1866
0
0.1700
0.0781
0.3789
0.3174
0.0046
…
…
…
…
…
…
…
…
…
…
495
0
0.0036
0.0406
0.2286
0.0600
0.0172
0
0.9387
0.0182
496
0.1588
0.0035
0.0107
1.4847
0.3163
0.0102
0
0
0.0016
497
0.1551
0.0009
0.0065
0
0.1453
0
0.7777
0
0.0033
498
0.1576
0.0011
0.0073
0
0.1679
0
0.5334
0
0.0060
499
0.1197
0.0006
0.0059
1.9414
0.2391
0.0172
0.4405
0.1432
0.0097
500
0.3600
0.0001
0.0049
1.9554
0.3960
0.0079
0.6331
0.1410
0.0093
501
0.0514
0.0033
0.0049
2.0436
0.3499
0.0111
0.7690
0.1376
0.0089
502
0.2503
0.0034
0.0042
2.0837
0.1744
0.0132
0.6198
0.1274
0.0066
503
0.1195
0.0147
0.0165
0
0.4283
0.0060
1.1613
0.1530
0.0049
504
0.0974
0.0144
0.0027
0
0.2241
0.0048
0.4778
0.0574
0.0043
505
0.1422
0.0060
0.0131
0
0.2023
0.0054
0.5604
0.0669
0.0004
…
…
…
…
…
…
…
…
…
…
1102
0.1898
0.0016
0.0413
0.8313
0.0767
0.0381
0.6875
0
0.1451
1103
0.2094
0.0015
0.0352
0.8262
0.0673
0.0451
0.7298
0.2808
0.0147
1104
0.1711
0.0011
0.0040
0.6648
0.0915
0.0462
0.5563
0.1831
0.0105
1105
0.1816
0.0012
0.0116
0.6658
0.1049
0.0371
0.6591
0.2874
0.0057
1106
0.1191
0.0011
0.0116
0.6260
0.1155
0.0393
0
0.0196
0.0060
1107
0.0417
0.0012
0.0041
0.4992
0.0781
0.0382
0
0
0.0065
1108
0.1058
0.0033
0.0045
0.0019
0.0589
0.0421
0
0
0.0018
1109
0.1740
0.0021
0
0
0.0446
0.0316
2.1187
0
0.4511
1110
0.2912
0.0021
0.0163
0.0385
0.0342
0.0037
0
1.1563
0.0257Figures6(a)–6(i) are the graphs of σ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ1,1m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively.Figure 6
The graph ofσ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ1,1m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Table5 shows the real and simulated estimates for the spot price processes pjt, j∈I1,n.Table 5
Real and simulated estimates (without jump) for natural gas, crude oil, and coal.
tk
Real
Natural gas
Real
Crude oil
Real
Coal
Simulatedp1sm^k,k
Simulatedp2sm^k,k
Simulatedp3sm^k,k
11
4.0200
4.0500
58.9900
56.5200
16.5900
16.8000
12
3.9900
4.0500
59.5200
59.3099
17.4600
16.8635
13
3.7500
3.6690
61.4500
59.3377
17.8900
17.8086
14
3.7700
3.6341
60.4900
59.4191
17.5500
17.0859
15
3.4100
3.3967
61.1500
59.6974
17.4100
17.0859
16
3.3500
3.3967
62.4800
59.6974
16.7500
17.0859
17
3.4900
3.4537
63.4100
61.2177
17.6600
19.0677
18
3.5500
3.4537
65.0900
61.4561
17.5200
16.0578
19
3.9200
3.8618
66.3100
61.6529
18.5000
19.0677
20
3.8600
3.8618
68.5900
60.9364
19.0600
19.0677
…
…
…
…
…
…
…
495
4.1900
4.0368
107.1800
104.1295
32.7600
31.3108
496
4.3300
4.1868
110.8400
111.1245
33.6500
32.7737
497
4.3300
4.1025
111.7200
112.4675
33.7100
33.4888
498
4.3700
4.0964
111.6800
110.8795
34.7500
35.5907
499
4.3200
4.1042
111.7200
104.2465
34.5400
32.9391
500
4.3500
4.0548
112.3100
109.9535
34.0400
36.2674
501
4.3800
4.0548
112.3800
109.9995
33.1000
36.2674
502
4.5100
4.3249
113.3900
104.3254
33.6700
34.8915
503
4.6000
4.3555
113.0300
113.2356
33.9400
35.0472
504
4.6000
4.3491
110.6000
103.9435
33.8300
32.8992
505
4.5900
4.3609
108.7900
104.9995
32.0200
32.8992
…
…
…
…
…
…
…
1102
3.7200
3.5963
108.2300
110.5149
4.7700
2.8861
1103
3.7300
3.5963
106.2600
105.8076
5.0100
5.6871
1104
3.6800
3.4099
104.7000
105.8076
4.9800
5.3821
1105
3.6600
3.4356
103.6200
105.8076
4.7300
4.9221
1106
3.5900
3.4636
103.2200
106.9547
4.6800
4.2352
1107
3.5200
3.2573
102.6800
105.4047
4.6300
5.8172
1108
3.4900
2.8981
103.1000
102.4928
4.7400
6.0376
1109
3.5100
2.8981
102.8600
102.4928
4.3300
5.1121
1110
3.4800
3.0267
102.3600
102.4928
4.1800
4.8978Figure7 below shows the graph of the real and simulated prices for natural gas, crude oil, and coal data set.Figure 7
Real and simulated prices (without jump) for natural gas, crude oil, and coal.
(a)
(b)
(c)Figures7(a)–7(c) show the graph of the real and simulated spot prices for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively. The red line represents the real data set ptk, while the blue line represent the simulated data set psm^k,k. Here, we begin by using a starting delay of r=10. The simulation starts from tr=t10. It is clear that the graph fits well. To reduce magnitude of error, we increase the magnitude of time delay. We later compare this result with the case where jump is incorporated into the system.


### 7.5.2. Relationship between Natural Gas, Crude Oil, and Coal: With Jump Incorporated

In this subsubsection, we analyze the relationship between natural gas, crude oil, and coal with the jump process. Here, we apply the above conceptual computational algorithm described in this section for the real-time data sets, namely, daily Henry Hub natural gas data set, daily crude oil data set, and daily coal data set for the period of 05/04/2009-01/03/2014, [35–37]. For i∈I1,K∗, K≠0, we use Δti−1=1, ε=0.001, ri−1=10, and di−1=2. The ε-level suboptimal estimates of the parameters at each real data times are described below for each commodity data sets.The parameters corresponding to the model governing natural gas price data set areu1i−1m^k,k, β1i−1m^ki−1,tki−1, κ1,1i−1m^ki−1,tki−1, κ1,2i−1m^ki−1,tki−1, κ1,3i−1m^ki−1,tki−1, γ1,1i−1m^ki−1,tki−1, γ1,2i−1m^ki−1,tki−1, γ1,3i−1m^ki−1,tki−1, δ1,1i−1m^ki−1,tki−1, δ1,2i−1m^ki−1,tki−1, δ1,3i−1m^ki−1,tki−1, σ1,1i−1m^ki−1,tki−1, σ1,2i−1m^ki−1,tki−1, and σ1,3i−1m^ki−1,tki−1. The parameters corresponding to the model governing crude oil price data set are u2i−1m^ki−1,tki−1, β2i−1m^ki−1,tki−1, κ2,1i−1m^ki−1,tki−1, κ2,2i−1m^ki−1,tki−1, κ2,3i−1m^ki−1,tki−1, γ2,1i−1m^ki−1,tki−1, γ2,2i−1m^ki−1,tki−1, γ2,3i−1m^ki−1,tki−1, δ2,1i−1m^ki−1,tki−1, δ2,2i−1m^ki−1,tki−1, δ2,3i−1m^ki−1,tki−1, σ2,1i−1m^ki−1,tki−1, σ2,2i−1m^ki−1,tki−1, and σ2,3i−1m^ki−1,tki−1, while the parameters corresponding to the model governing coal price data set are u3i−1m^ki−1,tki−1, β3i−1m^ki−1,tki−1, κ3,1i−1m^ki−1,tki−1, κ3,2i−1m^ki−1,tki−1, κ3,3i−1m^ki−1,tki−1, γ3,1i−1m^ki−1,tki−1, γ3,2i−1m^ki−1,tki−1, γ3,3i−1m^ki−1,tki−1, δ3,1i−1m^ki−1,tki−1, δ3,2i−1m^ki−1,tki−1, δ3,3i−1m^ki−1,tki−1, σ3,1i−1m^ki−1,tki−1, σ3,2i−1m^ki−1,tki−1, and σ3,3i−1m^ki−1,tki−1.For the sake of simplicity and in order to be able to compare our results in this subsection with the results in Subsection7.5.1, for each j,l∈I1,n, we rewrite the parameters uji−1m^ki−1,tki−1, βji−1m^ki−1,tki−1, κj,li−1m^ki−1,tki−1, γj,li−1m^ki−1,tki−1, δj,li−1m^ki−1,tki−1, and σj,li−1m^ki−1,tki−1 after they have been estimated as ujm^k,k, βjm^k,k, κj,lm^k,k, γj,lm^k,k, δj,lm^k,k, and σj,lm^k,k.We give some results for the jump times of the systemTii∈I1,K∗.Table6 shows some result for the jump times of the system (y,p). These results are derived by recording the times at which an entry of a commodity differ by twice the standard deviation or more from the mean of that commodity. These times are now combined into a single array and sorted out in an increasing order.Table 6
Some result for the jump times of the system (y,p).
T
17
44
61
87
157
200
422
464
483
502
722
754
870
930
1113We give the estimate of the jump coefficient matricesΠi and Θi defined in (34) in Table 7 below.Table 7
Estimatesπ1i, π2i, π3i, θ1i, θ2i, and θ3i.
i
Ti
Natural gas
Crude oil
Coal
Ti
Natural gas
Crude oil
Coal
Ti
Π1i
Π2i
Π3i
Ti
θ1i
θ2i
θ3i
1
17
1.0031
1.1219
1.0256
17
1.0049
1.1219
1.0493
2
44
0.9213
0.9727
1.0410
44
0.9352
1.0084
0.9249
3
61
0.9482
0.9671
0.9661
61
0.9997
0.9427
0.9404
4
87
0.8859
0.9974
0.9653
87
0.7389
1.0452
0.9905
5
157
1.0435
0.9350
1.0432
157
1.0933
1.0019
1.0049
6
200
1.0309
1.0199
1.0382
200
0.9826
1.0210
0.9794
7
422
1.0270
0.9775
0.9669
422
0.9706
0.9939
0.9917
8
464
0.9581
1.0462
1.0523
464
1.0128
1.0508
1.0324
9
483
0.9765
0.9787
1.0291
483
1.0382
1.0328
1.0246
10
502
1.0532
1.0737
1.0136
502
1.0359
1.0073
1.0162
11
722
0.9812
0.9959
0.9919
722
0.9700
0.9695
1.0011
12
754
1.0003
1.0009
0.9189
754
1.0137
0.9987
1.3481
13
870
1.0579
0.9921
1.1378
870
1.0328
1.0033
1.1420
14
930
1.0275
0.9907
0.9978
930
0.9995
0.9812
1.1848
15
1113
1.0009
0.9960
1.0706
1113
0.9304
0.9801
0.9897Table8 gives the estimates u1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for the decoupled dynamical system for y with jump.Table 8
Estimatesm^k, u1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, m^k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, m^k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k.
tk
m^k
Natural gas
Crude oil
Coal
u1m^k,k
κ1,1m^k,k
κ1,2m^k,k
κ1,3m^k,k
u2m^k,k
κ2,1m^k,k
κ2,2m^k,k
κ2,3m^k,k
u3m^k,k
κ3,1m^k,k
κ3,2m^k,k
κ3,3m^k,k
×10−16
×10−16
×10−16
×10−16
×10−16
×10−16
11
1
4.1593
0
0
0
57.7000
0
0
0
16.7407
0
0
0
12
4
4.2000
0
0
0
58.6313
0.0011
0.0310
-0.0012
16.2395
0
0
-0.0376
13
6
4.0616
0.0679
-0.0054
-0.0035
58.5378
-0.0035
0.0205
0.0032
16.2680
0
0
0.1069
14
2
4.0616
-0.0242
-0.0179
-0.0035
61.4809
0.0020
0.0098
0
15.5249
0
0
-0.0294
15
7
4.0910
0.6416
-0.2898
0.0078
22.5758
-0.0057
0.0085
0.0012
18.7073
0.0009
-0.0021
0.0318
16
4
4.0160
0
0
0.0078
59.6867
-0.0051
0.0080
0
17.0060
0
-0.0021
0
17
2
4.9575
0
0
0.0078
60.3710
-0.0005
0.0207
0
12.8918
-0.0005
-0.0002
0.0318
18
8
4.9575
-0.1947
0
0.0078
62.3437
0.0005
-0.0008
0
16.5954
0.0002
0.0008
0.0662
19
4
3.3190
-0.4472
0.6760
0.0078
74.6911
-0.0008
-0.0019
0
17.9932
0
-0.0002
0
20
1
3.4762
-0.2540
-0.0048
0.0078
65.9190
0.0026
-0.0006
1
17.6485
0
0
0.0499
…
…
…
…
…
…
…
…
…
…
…
…
…
…
494
1
4.1457
0
0
-0.0001
115.1875
0.0002
0.0053
0
33.3359
0.0003
-0.0003
0.0326
495
1
4.2877
0.1184
-0.0014
0
124.5218
0.0008
0.0056
0
30.1732
0.0002
-0
0.0412
496
1
4.2238
0.2582
0.0011
0.0003
106.8349
0.0003
0.0113
-0.0006
34.9907
0.0034
0
0.0097
497
5
4.0998
0.0477
-0.0006
-0.0002
108.4725
-0.0003
0.0162
-0.0033
33.3388
0.0002
0
0.0443
498
8
4.0592
0.0201
0.0010
0
104.8926
0
0
0.0003
35.1174
0
0.0001
0.0207
499
1
4.3433
0.2118
-0.0014
0
109.2551
-0.0002
0.0048
0.0003
33.2862
0.0010
0.0001
0.0068
500
4
2.4519
0
0
0
111.7067
0
0
0
36.1647
0.0003
0.0003
0.0079
501
1
4.2415
0.4108
0
-0.0015
110.7517
-0.0006
0.0026
0.0009
34.9145
-0.0001
-0.0004
0.0407
502
2
4.3633
0.3210
0.0002
-0.0001
103.6326
-0.0019
0.0023
-0.0002
34.8337
-0.0001
-0.0001
0.0140
503
2
4.2911
0.1276
0.0003
0.0043
112.1547
-0.0033
0.0030
0.0027
35.8389
0.0005
0.0001
0.0211
504
7
4.5942
-0.0125
-0.0002
-0.0031
111.1278
0.0010
0.0072
0.0006
33.6875
-0.0021
0
0.0268
505
2
3.1882
0.0666
0.0009
0
106.1919
-0.0009
0.0110
0.0011
33.6640
-0.0011
-0.0002
0.0231
…
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
1
3.5909
0
0
0.0008
110.3777
0.0006
0.0045
0
5.1761
0.0067
-0.0029
-0.0044
1103
6
3.5303
0.1166
0.0002
-0.0003
111.1585
-0.0003
0.0083
0
5.4558
-0.0019
0.0014
0.0600
1104
4
3.5314
0.0809
0.0018
0
109.0996
-0.0007
0.0095
0.0013
4.8000
0.0005
0.0006
0.1742
1105
1
3.7100
0.2234
-0.0013
-0.0015
106.5667
0.0033
0.0073
-0.0020
5.4226
-0.0082
0.0020
0.0932
1106
8
3.4084
0.1098
0.0001
0
106.5989
0.0003
0.0030
0.0023
5.3360
-0.0023
0.0005
0.0956
1107
7
3.5520
0.1086
0.0001
-0.0070
103.4473
-0.0020
0.0037
-0.0045
4.3586
-0.0005
0.0004
0.1418
1108
5
3.9233
0.0601
0.0007
0
102.8550
0
0.0040
0
4.6582
-0.0010
0
0.1388
1109
8
3.5328
0.0417
0
0
103
-0.0002
0.0089
-0.0005
4.9663
-0.0019
0.0008
0.1279
1110
5
3.8399
0.0212
0.0004
0
102.8800
0
0
0
4.7286
-0.0037
-0.0030
0.0740Figure8 shows the parameter estimates u1m^k,k, u2m^k,k, and u3m^k,k for the decoupled dynamical system for y with jump.Figure 8
The graph of mean levelu1m^k,k, u2m^k,k, and u3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)Figures8(a)–8(c) are the graphs of u1m^k,k, u2m^k,k, and u3m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively. By plotting the real data sets, it is easily seen that the graphs of u1m^k,k, u2m^k,k, and u3m^k,k are similar to the graph of the Henry Hub natural gas, crude oil, and coal data set, respectively. We expect this to happen because uj, j∈I1,3, are the equilibrium spot price processes of (5).The graph of the interaction parametersκ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for the decoupled dynamical system for y with jump and estimates in Table 8 are given in Figure 9 below.Figure 9
The graph of interaction coefficientsκ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures9(a)–9(i)) show the graph of the ε suboptimal interaction coefficient parameters κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k. The interaction coefficients κj,l, j≠l, are negligible, because each estimate is <<10−15. Thus, this shows that the model describing the mean spot price, yj, is mainly characterized by the market potential κj,ji−1uji−1−yjyj, j∈I1,n, i∈I1,K∗.Table9 below shows the estimates of the diffussion coefficient’s parameters δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k,δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k for the decoupled dynamical system for y with jump.Table 9
Estimatesδ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
δ1,1m^k,k
δ1,2m^k,k
δ1,3m^k,k
δ2,1m^k,k
δ2,2m^k,k
δ2,3m^k,k
δ3,1m^k,k
δ3,2m^k,k
δ3,3m^k,k
11
0.0062
0.0010
0.0001
1.5277
0.0078
0.0011
0
0
0.0218
12
0.0182
0.9002
0
1.6227
0.0010
0
0
0
0.0988
13
0.0239
0.0802
1.7280
0
1.7694
0
0.6374
0.6374
0.0959
14
0
0.0001
0.6027
2.3258
0
0
1.6564
1.6564
0.0847
15
0
0.8001
0.6210
2.3252
0
0
1.6650
1.6650
0.0111
16
0.0455
0.0007
3.6877
2.3217
0
1.2215
1.6724
1.6724
0
17
0
0.9876
0
1.6425
0
0
1.7719
1.7719
0
18
3.0410
0.9351
0
1.3105
0
0.1070
1.7630
1.7630
0.0434
19
2.7713
0.6680
0
1.1052
0
0
1.7400
1.7400
0
20
2.8461
1.7795
0
0.1196
0
0.0983
0
0.4555
0
…
…
…
…
…
…
…
…
…
…
495
1.1229
0
0.0584
0.5488
0.1104
0.0761
0
0
1.3987
496
0.6946
0
0.6613
0.5767
0.0715
0.0610
0
0
1.3017
497
1.1229
0.0095
0.0988
0.6499
0.0870
0.0633
1.1317
1.1317
1.3069
498
0.6946
0.0101
0
0
0
0.0320
1.0294
1.0294
1.5410
499
0.7353
0.0066
0.0384
0
0.0922
0.0330
0.7317
0.7317
1.2225
500
1.7509
0.0069
0.0283
0.4307
0.4545
0.0413
0.4826
0.4826
1.2254
501
2.1299
0.0077
0.0282
0.5043
0.7873
0.0308
0.4272
0.4272
1.5587
502
0.9778
0.0077
0
0.2878
0
0
0.5239
0.5239
1.8713
503
0.9872
0
0
0.2909
0
0
1.4523
1.4523
1.8874
504
1.1329
0
0
0.3707
0.4261
0
0
0
0
505
1.9178
0
0
0.3812
0.7292
0.1724
0
0
0
…
…
…
…
…
…
…
…
…
…
1102
0
0.0331
0.7183
0.9297
0.0434
0.0680
0
0
1.1355
1103
1.5077
0.0626
0.2048
1.1017
0.0421
0.1510
0
0
1.4133
1104
0.4444
0.0435
0.4622
0.1939
0.1078
0
0.0814
0
1.1672
1105
3.5933
0
0.3646
0.1922
0
0.7273
0.2726
0.2726
1.3023
1106
2.4964
0
0.3919
0
0.0684
1.0179
0.3296
0.3296
1.4111
1107
2.4600
0
0.8995
0.2001
0.1510
0.9354
0
0
1.7245
1108
2.0262
0
0.6325
0.3781
0.0814
0.8825
0.1878
0.1878
1.0915
1109
1.7828
0
0.6116
0.4024
0.0332
0.8812
0
0
1.3191
1110
1.2706
0
0.1001
0.3252
0.0155
0.8078
0
0
1.0233The graph of the diffusion coefficient’s parameter for the decoupled dynamical system fory with jump are given in Figure 10 below.Figure 10
The graph of interaction coefficientsδ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures10(a)–10(i) show the graph of the ε- sub-optimal interaction measure of fluctuation coefficient parameters δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.Table10 gives the drift coefficient’s parameter estimates β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k for the dynamical system for p with jump.Table 10
Estimatesβ1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k.
tK
Natural gas
Crude oil
Coal
β1m^k,k
γ1,1m^k,k
γ1,2m^k,k
γ1,3m^k,k
β2m^k,k
γ2,1m^k,k
γ2,2m^k,k
γ2,3m^k,k
β3m^k,k
γ3,1m^k,k
γ3,2m^k,k
γ3,3m^k,k
11
0
0
0
0
0
0
0
0
0
0
0
0
12
0.1681
0.3497
-0.0109
0.0248
-0.4815
0.1626
-0.4066
-0.0123
0.7665
-0.3259
0.0205
-0.0198
13
0.1592
0.3755
-0.0102
0.0228
-0.7778
-0.5752
-0.0578
0.1870
1.0795
-0.2904
0.0135
-0.0217
14
7.3439
0.3755
0.0488
-0.5478
1.7680
-0.5555
-0.2058
0.0291
0.8543
-0.2056
0.0034
-0.0064
15
0.3336
0.3652
-0.0127
0.0213
-0.8999
0.0601
-0.1110
0.0405
0.5144
-0.1264
-0.0062
0.0127
16
0.4709
0.2780
-0.0116
0.0104
-0.8999
0.0601
-0.1110
-0.0292
0.0017
-0.0002
0
0
17
0.3277
0.2780
0.0768
-0.2633
1.3349
0.0027
-0.0330
-0.0750
-0.6285
-0.0569
0.0016
0.0262
18
0.3277
1.3156
-0.1491
-0.1646
1.5419
-0.0088
0.1205
-0.0892
-1.3275
-0.0703
0.0080
0.0386
19
0
0
0
0
-0.1785
-0.0368
-0.0062
0.0189
-0.8091
-0.0001
-0.0083
0.0466
20
0.6985
0.4990
-0.0069
-0.0187
-0.1513
-0.0778
-0.0096
0.0255
-0.0182
0.0080
0.0001
0.0003
…
…
…
…
…
…
…
…
…
…
…
…
…
495
0.1201
0.2264
0.0007
-0.0056
0.2756
0.1131
-0.9587
-0.0087
-0.0288
-0.0780
-0.0019
0.0136
496
0.1809
0.2085
0.0009
-0.0082
0.2898
-0.0431
0.7329
-0.0133
0.1324
-0.1434
-0.0018
0.0158
497
0.2442
0.1597
0.0007
-0.0093
3.1030
-0.0495
-0.0462
-0.0862
0.9772
-0.2426
-0.0065
0.0177
498
0.2742
0.2651
0.0020
-0.0145
1.3147
0.0148
-0.0009
-0.0411
0.2770
-0.1888
-0.0037
0.0217
499
0.3320
0.3298
-0.0009
-0.0070
0.8430
0.0070
0.0283
-0.0265
0.0931
-0.1551
-0.0041
0.0237
500
0.5035
0.2337
-0.0007
-0.0128
1.3320
0.3949
-0.3308
-0.0838
0.4175
-0.2331
-0.0034
0.0222
501
0.6328
0.2612
-0.0034
-0.0077
0.3251
-0.1148
0.0548
0.0042
0.7896
-0.2546
-0.0093
0.0305
502
0.5403
0.2457
-0.0014
-0.0113
0.4863
-0.1315
0.0343
0.0020
3.7990
0.0420
-0.0524
0.0421
503
0.4794
0.2098
-0.0028
-0.0050
0.1239
0.0013
0.0202
-0.0040
9.6735
0.0736
-0.1152
0.0648
504
-0.3308
-0.5600
0.0258
-0.0737
0.2867
-0.0391
0.0009
-0.0034
4.2547
0.3506
-0.0669
0.0382
505
1.1680
0.8346
-0.0274
0.0542
0.1198
-0.0588
-0.3412
0.0092
2.2295
0.0897
-0.0357
0.0306
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
0.6765
0.0455
-0.0020
-0.0908
0.4026
-0.2544
0.2045
0.1058
-5.9294
0.6777
0.0292
0.0068
1103
1.1804
0.4214
-0.0149
0.0837
-0.6549
0.1780
0.0070
0.0018
-6.3380
0.7440
0.0291
0.0106
1104
0.1069
0.2489
-0.0009
-0.0014
-2.1178
0.3406
0.1959
0.1826
-3.8701
0.5681
0.0157
0.0021
1105
0.0139
0.2777
-0.0001
-0.0008
0.3958
-0.0274
0.0642
-0.0620
-4.0701
0.1880
0.0091
0.0514
1106
-0.2513
0.4043
0.0031
-0.0164
0.4097
0.0060
0.1536
-0.0907
-5.0668
0.3261
0.0178
0.0419
1107
0.0670
0.3163
-0
-0.0145
0.2906
0.0485
0.2310
-0.0989
-5.0668
0.4016
0.0308
0.1474
1108
1.0112
0.6861
-0.0091
-0.0107
0.4281
0.0048
0.1337
-0.0933
-5.0668
0.4650
0.0304
0.1295
1109
0.5020
0.5370
-0.0030
-0.0375
0.3645
-0.0168
0.1078
-0.0641
-5.0668
0.4156
0.0311
0.1396
1110
0.1420
0.3295
0.0009
-0.0484
0.1728
-0.0189
-0.0164
-0.0230
-6.6650
0.4509
0.0099
0.0831Table10 shows the estimates of the ε suboptimal size m^k and the parameters m^k,β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, m^k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, m^k,β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k, for each of the energy commodity data sets. According to (58), the estimate γj,lm^k,k, j≠l, is positive if commodity pl is cooperating with commodity pj, and negative if commodity pl is competing with commodity pj. There is no interaction between the two commodities if γj,lm^k,k=0. It is apparent from the graph (from γ1,3m^k,k in Column 6) that coal is competing with natural gas during this period because the estimates of γ1,3m^k,k are mostly negative. It is apparent that natural gas and crude oil are either cooperating or competing, depending on the time period.The graph of the drift coefficient’s parameters with estimates in Table10 are given in Figure 11 below.Figure 11
The graph of interaction coefficientsγ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures11(a)–11(i) show the graph of the ε suboptimal interaction coefficient parameters γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k. According to (58), the estimate γj,lm^k,k, j≠l, is positive if commodity pl is cooperating with commodity pj, and negative if commodity pl is competing with commodity pj. There is no interaction between the two commodities if γj,lm^k,k=0. It is apparent from the graph of γ1,3m^k,k that coal is competing with natural gas because the estimates of γ1,3m^k,k are mostly negative. This shows that for fixed price p1 of natural gas, for every $1 increase in the price of coal, the price of natural gas decreases on the average by 0.0019. Also, it is apparent that natural gas and crude oil are either cooperating or competing, depending on the time period.Figures12(a)–12(c) are the graphs of β1m^k,k, β2m^k,k, and β3m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively.Figure 12
The graph ofβ1m^k,k, β2m^k,k, and β3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)Table11 gives the ε suboptimal estimates of the parameters u1m^k,k, σ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, u2m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, u3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k for each of the energy commodity data sets.Table 11
Estimatesσ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
σ1,1m^k,k
σ1,2m^k,k
σ1,3m^k,k
σ2,1m^k,k
σ2,2m^k,k
σ2,3m^k,k
σ3,1m^k,k
σ3,2m^k,k
σ3,3m^k,k
11
0
0
0
0
0
0
0
0.1303
0
12
0.0485
0.0004
0.0032
0.2734
0.0166
0
0.0513
0
0.0000
13
0.7333
0
0
0.9445
0
0
0.2489
0
0
14
0.2120
0.1386
0.0133
0.3877
0.4773
0.1195
0.1365
0.0665
0.0086
15
0.4246
0.1318
0.0021
0.03341
0.4894
0.1211
0.0112
0.6107
0.0696
16
0.5538
0.0778
0.1501
0.07751
0.2524
0.0811
0.0651
0.4251
0.0635
17
0.3907
0.0469
0.2230
0.08746
0.1848
0.2463
0
0.4458
0.0478
18
0.3523
0.0180
0.2178
0.04291
0.1877
0.1602
0.5681
0.0592
0.0115
19
0.5116
0.0619
0.2221
0.03266
0.2673
0.2465
0.4999
0.0569
0.0127
20
0.6431
0.0536
0.1866
0.0939
0.1700
0.0781
0.3789
0.3174
0.0046
…
…
…
…
…
…
…
…
…
…
495
0
0.0036
0.0406
0.2286
0.0600
0.0172
0.0110
0.9387
0.0182
496
0.1588
0.0035
0.0107
0.08183
0.3163
0.0102
0
0
0.0016
497
0.1551
0.0009
0.0065
0.07869
0.1453
0.4821
0.7777
0
0.0033
498
0.1576
0.0011
0.0073
0.0120
0.1679
0.3786
0.5334
0
0.0060
499
0.1197
0.0006
0.0059
0.0721
0.2391
0.0172
0.4405
0.1432
0.0097
500
0.3600
0.0001
0.0049
0.0273
0.3960
0.0079
0.6331
0.1410
0.0093
0.5010
0.0514
0.0033
0.0049
0.0182
0.3499
0.0111
0.7690
0.1376
0.0089
0.5020
0.2503
0.0034
0.0042
0.0222
0.1744
0.0132
0.6198
0.1274
0.0066
0.5030
0.1195
0.0147
0.0165
0
0.4283
0.0060
0
0.1530
0.0049
0.5040
0.0974
0.0144
0.0027
0
0.2241
0.0048
0.4778
0.0574
0.0043
0.5050
0.1422
0.0060
0.0131
0.0085
0.2023
0.0054
0.5604
0.0669
0.0004
…
…
…
…
…
…
…
…
…
…
1102
0.1898
0.0016
0.0413
0.8313
0.0767
0.0381
0.6875
0
0.1451
1103
0.2094
0.0015
0.0352
0.8262
0.0673
0.0451
0.7298
0.2808
0.0147
1104
0.1711
0.0011
0.0040
0.6648
0.0915
0.0462
0.5563
0.1831
0.0105
1105
0.1816
0.0012
0.0116
0.6658
0.1049
0.0371
0.6591
0.2874
0.0057
1106
0.1191
0.0011
0.0116
0.6260
0.1155
0.0393
0
0.0196
0.0060
1107
0.0417
0.0012
0.0041
0.4992
0.0781
0.0382
0.0271
0.2559
0.0065
1108
0.1058
0.0033
0.0045
0.0019
0.0589
0.0421
0.6209
0.8289
0.0018
1109
0.1740
0.0021
0
0.0305
0.0446
0.0316
0.8431
0.2366
0.4511
1110
0.2912
0.0021
0.0163
0.0385
0.0342
0.0037
0.2910
0.0489
0.0257Figures13(a)–13(c) are the graphs of σ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively.Figure 13
The graph ofσ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Table12 shows the real and simulated estimates for the spot price processes pjt, j∈I1,n.Table 12
Real and simulated estimates (with jump) for natural gas, crude oil, and coal.
tk
Natural gas
Crude oil
Coal
Real
Simulatedp1sm^k,k
Real
Simulatedp2sm^k,k
Real
Simulatedp3sm^k,k
11
4.0200
4.0500
58.9900
58.5200
16.5900
16.6000
12
3.9900
3.9600
59.5200
59.5099
17.4600
17.4635
13
3.7500
3.6690
61.4500
60.3377
17.8900
17.8886
14
3.7700
3.7341
60.4900
59.4191
17.5500
17.5188
15
3.4100
3.3967
61.1500
60.1580
17.4100
17.4188
16
3.3500
3.3947
62.4800
62.5028
16.7500
16.7789
17
3.4900
3.3900
63.4100
63.5524
17.6600
17.5341
18
3.5500
3.4957
65.0900
64.9224
17.5200
17.8130
19
3.9200
3.8743
66.3100
66.8239
18.5000
18.8453
20
3.8600
3.8241
68.5900
68.1206
19.0600
18.9453
…
…
…
…
…
…
…
494
4.2300
4.2295
106.7000
106.6374
33.2200
33.1852
495
4.1900
4.2191
107.1800
106.9973
32.7600
32.6677
496
4.3300
4.3414
110.8400
111.0084
33.6500
33.8070
497
4.3300
4.3261
111.7200
111.7084
33.7100
33.7061
498
4.3700
4.3863
111.6800
111.2084
34.7500
35.7057
499
4.3200
4.2167
111.7200
111.6126
34.5400
33.5457
500
4.3500
4.3577
112.3100
112.4358
34.0400
34.2862
501
4.3800
4.3535
112.3800
112.5682
33.1000
33.1330
502
4.5100
4.4389
113.3900
113.2925
33.6700
33.6216
503
4.6000
4.6139
113.0300
113.3077
33.9400
33.9216
504
4.6000
4.5964
110.6000
110.8350
33.8300
33.9216
505
4.5900
4.5564
108.7900
108.7598
32.0200
31.9867
…
…
…
…
…
…
…
1102
3.7200
3.6616
108.2300
108.2354
4.7700
4.5482
1103
3.7300
3.6477
106.2600
106.1451
5.0100
5.1120
1104
3.6800
3.6748
104.7000
104.6723
4.9800
5.1180
1105
3.6600
3.6861
103.6200
104.6723
4.7300
4.7893
1106
3.5900
3.6436
103.2200
102.9765
4.6800
4.7074
1107
3.5200
3.5213
102.6800
102.7652
4.6300
4.6419
1108
3.4900
3.4564
103.1000
102.9765
4.7400
4.4016
1109
3.5100
3.2596
102.8600
102.9652
4.3300
4.1826
1110
3.4800
3.4604
102.3600
102.4345
4.1800
4.4606Figure14 shows the graph of the real and simulated prices (with jump) for natural gas, crude oil, and coal data set.Figure 14
Real and simulated prices (with jump) for natural gas, crude oil, and coal.
(a)
(b)
(c)Figures14(a)–14(c) show the graph of the real and simulated spot prices for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively. The red line represents the real data set p, while the blue line represent the simulated data set pm^k,ks. It is clear that the graph fits well. To reduce magnitude of error, we increase the magnitude of time delay. It is obvious that these curves fit better than the curves in Figure 7. It follows that the interconnected dynamical system with jump process incorporated into it performs better than the one without jump.


## 7.5.1. Illustration: Relationship between Natural Gas, Crude Oil, and Coal: Without Incorporating Jump Process

In this subsubsection, we analyze the relationship between natural gas, crude oil, and coal without the jump process. Forj,l∈I1,3, the stochastic dynamic system governing the three energy commodities is described in (59) of Remark 12. Here, (y1,p1) denotes the mean spot and the spot price process of natural gas, (y2,p2) denotes the mean spot and the spot price process of crude oil, and (y3,p3) denotes the mean spot and the spot price process of coal.Using the discretized scheme (82), we apply the above conceptual computational algorithm for the real-time data sets namely daily Henry Hub natural gas data set, daily crude oil data set, and daily coal data set for the period of 05/04/2009-01/03/2014, [35–37]. Using Δt=1, ε=0.001, r=10, and d=2, the ε-level suboptimal estimates of the parameters at each real data times are described below for each commodity data sets.The parameters corresponding to the natural gas data set areu1m^k,k, β1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, σ1,1m^k,k, σ1,2m^k,k, and σ1,3m^k,k. The parameters corresponding to the crude oil data set are u2m^k,k, β2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, and σ2,3m^k,k. The parameters corresponding to coal data set are u3m^k,k, β3m^k,p3,k, κ3,1m^k,k, κ3,2m^k,k, κ3,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, γ3,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, δ3,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k.The following table gives the drift coefficient’s parameter estimatesu1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for the decoupled dynamical system for y in the case where jump is not incorporated into the dynamical system.Table1 shows the estimates of the ε suboptimal size m^k, j∈I1,3, the parameters u1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for each of the energy commodity data sets. Moreover, d≤r and the initial real data time is tr=t10.Table 1
Estimatesm^k, u1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k.
tk
m^k
Natural gas
Crude oil
Coal
u1m^k,k
κ1,1m^k,k
κ1,2m^k,k
κ1,3m^k,k
u2m^k,k
κ2,1m^k,k
κ2,2m^k,k
κ2,3m^k,k
u3m^k,k
κ3,1m^k,k
κ3,2m^k,k
κ3,3m^k,k
×10−16
×10−18
×10−18
×10−18
×10−18
×10−18
11
1
4.1593
0.0211
0
0
57.7000
0
0
0
16.7407
0
0
0
12
3
4.2000
0.0111
0
0
58.6313
0.0011
0.0310
-0.0012
16.2395
0
0
-0.0376
13
5
4.0616
0.0679
-0.0054
-0.0035
58.5378
-0.0035
0.0205
0.0032
16.2680
0
0
0.1069
14
5
4.0616
-0.0242
-0.0179
0
61.4809
0.0020
0.0098
0
15.5249
0
0
-0.0294
15
8
4.0910
0.6416
-0.2898
0
58.9282
-0.0036
0.0128
0.0071
16.8286
0
0
0.0513
16
8
4.0160
0.2101
0
0
59.6867
-0.0051
0.0080
0.0071
17.0888
0
0
0.0415
17
8
4.9575
0.1876
0
0
60.6244
0.0024
0.0052
0
17.4120
-0.0003
0.0001
0.0555
18
8
4.9575
-0.1947
0
0
61.0700
0
0
0
17.2374
0
-0
0
19
6
4.7336
-1.4476
5.8820
0
61.9414
0
0.0043
-0.0086
16.8438
0.0001
0.0001
0.0768
20
6
2.5646
0.3319
0.7261
0
62.7899
0
0.0053
0.0082
18.3022
-0.0083
0.0027
0.0558
…
…
…
…
…
…
…
…
…
…
…
…
…
…
495
8
3.9654
0.0591
-0
0
108.2457
0.0038
0.0049
-0.0023
33.1313
0.0027
0.0009
0.0363
496
5
4.0421
0.0616
0.0001
0.0017
107.5186
0
0
0
33.4224
-0.0005
0.0003
0.0214
497
6
4.0514
0.0127
-0.0002
0.0020
109.8836
0
0
-0.0001
33.3388
0.0002
0
0.0443
498
7
4.1646
0.0442
-0.0012
-0.0053
107.8013
-0.0021
0.0033
0.0038
33.2862
-0.0002
0.0006
0.0343
499
6
4.1226
0.0352
-0.0020
0
108.1554
-0.0005
0.0032
0.0039
33.2862
0.0010
0.0001
0.0068
500
6
4.2625
0.0733
-0.0002
0
110.5101
-0.0032
0.0033
0.0016
36.1647
0.0003
0.0003
0.0079
501
8
3.1551
0
0
-0.0009
110.3071
0.0014
0.0025
0
34.7467
0
0
0
502
4
4.1564
0.0914
-0.0002
0
111.1186
0
0.0013
-0.0031
49.4050
0.0026
-0.0002
0.0211
503
5
4.5799
0.0467
0.0004
0
112.0057
0
0.0027
-0.0043
34.7207
-0.0001
-0.0001
0.0216
504
4
4.3061
0.0236
0.0002
0.0007
112.3186
0
0.0021
0.0015
34.4483
0.0019
0.0003
0.0170
505
9
4.4325
-0.0015
-0.0018
0.0030
106.3345
0
0.0043
0.0001
33.7160
0
-0.0006
0.0265
…
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
7
3.5429
-0.0286
-0.0006
-0.0028
110.3777
0.0006
0.0045
0
5.2399
0
0.0013
0.0008
1103
4
3.5601
0.1028
0.0001
0.0001
111.1585
-0.0003
0.0083
0
5.4824
0
0.0077
0.0485
1104
4
3.5314
0.0809
0.0018
0.0090
109.0996
-0.0007
0.0095
0.0013
11.0949
-0.0018
0.0005
0.1175
1105
4
3.4439
0.1551
-0.0008
-0.0015
106.5667
0.0033
0.0073
-0.0020
4.8300
-0.0012
-0.0003
0.1283
1106
6
3.8206
0.2258
0.0004
0
104.7497
0
0
0.0027
4.8300
0
0.0008
0
1107
4
3.6917
0.2132
-0.0001
-0.0008
105.1229
0.0011
0.0039
0
4.3586
-0.0005
0.0004
0.1418
1108
5
3.7871
0
0
0
105.3595
0.0006
0.0027
-0.0009
4.8000
0.0006
-0.0001
0.1265
1109
4
3.8445
-0.0405
-0.0011
0.0011
102.9022
-0.0044
0.0037
0.0039
5.0279
0
0
0
1110
5
3.8399
0.0212
0.0004
0
102.8313
-0.0020
0.0045
0.0018
4.6817
0.0021
0.0041
0.0536Figure1 shows the drift coefficient’s parameter estimates u1m^k,k, u2m^k,k, and u3m^k,k for the decoupled dynamical system for y in the case where jump is not incorporated into the dynamical system.Figure 1
The graph of mean levelu1m^k,k, u2m^k,k, and u3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)Figures1(a)–1(c) are the graphs of u1m^k,k, u2m^k,k, and u3m^k,k against time tk for the daily Henry Hub natural gas price [36], daily crude oil price [37], and daily coal price [35] data set, respectively. By plotting the real data sets, it is easily seen that the graphs of u1m^k,k, u2m^k,k, and u3m^k,k are similar to the graph of the Henry Hub natural gas, crude oil, and coal data set, respectively. We expect this to happen because uj, j∈I1,3, are the equilibrium spot price processes of (5). The sample means of the real spot price data sets for natural gas, crude oil, and coal are given by 3.7218, 88.5620, and 16.5543, respectively. It can be seen from Figures 1(a)–1(c) that the graphs of u1m^k,k, u2m^k,k, and u3m^k,k move around the mean values 3.7218, 88.5620, and 16.5543, respectively. This analysis shows that the parameters uj, j∈I1,3, are statistic processes for the respective mean of the data sets at time tk.The graph of the interaction parametersκ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for the decoupled dynamical system for y (with no jump incorporated into the dynamical system) are given in Figure 2 below.Figure 2
The graph of interaction coefficientsκ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures2(a)–2(i) show the graph of the ε suboptimal interaction coefficient parameters κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k. The interaction coefficients κj,l, j≠l are negligible, because each estimate is <<10−15. Thus, this shows that the model describing the mean spot price, yj, is mainly characterized by the market potential κj,juj−yjyj, j∈I1,n.Table2 below shows the estimates of the diffusion coefficient’s parameters δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k,δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k for the decoupled dynamical system for y in the case where jump is not incorporated into the dynamical system.Table 2
Estimatesδ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k,δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
δ1,1m^k,k
δ1,2m^k,k
δ1,3m^k,k
δ2,1m^k,k
δ2,2m^k,k
δ2,3m^k,k
δ3,1m^k,k
δ3,2m^k,k
δ3,3m^k,k
11
0.0123
0.0012
0.0001
0
0.0223
0
0.0412
0
0.0022
12
0.0024
0.0011
0.0121
0.0234
0.0245
0
0
0
0.0112
13
0.0001
1.3425
1.7280
1.9811
0.9899
0.9731
0.6374
0.6374
0.0123
14
0
1.1267
0.6027
2.3258
0.1213
3.9128
1.6564
1.6564
0.0004
15
1.15260
0.4287
0.6210
2.3252
0.0006
0.5083
1.6650
1.6650
0.4565
16
4.9354
0
0
2.3217
0.0120
1.1124
1.6724
1.6724
0.8762
17
4.1360
0.0989
3.6877
1.6425
0
0
1.7719
1.7719
0
18
3.0410
0.1527
0
1.3105
0.9167
1.3451
1.7630
1.7630
0
19
2.7713
0
0
1.1052
0
3.3241
1.7400
1.7400
0
20
2.8461
0.0012
0.2221
0.1196
5.1929
0
0.6532
0.9876
0.0082
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
494
2.9961
0.0586
0
0.5529
0
0.42339
0
0
0.5187
495
5.9059
0
0.0584
0.5488
0.8947
0
0.0017
0.0021
0.0001
496
0.1121
0
0.6613
0.5767
0.9899
0
0.8763
0
0.9827
497
1.1229
0.0095
0.0988
0.6499
5.8547
0
1.1317
1.1317
0.0012
498
0.6946
0.0101
0
0
5.8298
0.0320
1.0294
1.0294
0.0321
499
0.7353
0.0066
0.0384
0
5.7180
0.0330
0.7317
0.7317
0.0431
500
1.7509
0.0069
0.0283
0.4307
5.6133
0.0413
0.4826
0.4826
0.0783
501
2.1299
0.0077
0.0282
0.5043
5.6282
0.0308
0.4272
0.4272
0.0002
502
0.9778
0.0077
0.0255
0.2878
4.6543
0.0322
0.5239
0.5239
0.0098
503
0.9872
0
0
0.2909
4.5544
0.0411
1.4523
1.4523
0.0087
504
1.1329
0
0
0.3707
0
0.1128
2.4181
2.4181
0
505
1.9178
0
0
0.3812
1.3243
0.1724
4.9207
4.9207
0
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
0
0.0331
0.0056
0.9297
3.9502
0
0.2853
1.8033
1.1355
1103
1.5077
0.0626
0.0332
1.1017
2.8221
0
0
0
1.4133
1104
0
0.0435
0.5821
0.1939
4.5585
0
0
0
1.1672
1105
0
0
1.52970
0.1922
3.2418
0.7273
0.2726
0.2726
0
1106
4.4476
0.323
0.5112
3.5487
3.8113
1.0179
0.3296
0.3296
0
1107
2.4312
0.0011
0.0435
0.2001
2.6026
0.9354
0
0
1.7245
1108
2.5079
0.1232
0.4542
0.3781
0
0.8825
0.1878
0.1878
0
1109
1.7828
0.0431
0.3210
0.4024
0
0.8812
0
0
1.3191
1110
1.2706
0.0056
1.1123
0.3252
0
0.8078
0
0
1.0233The graph of the diffusion coefficient’s parameter for the decoupled dynamical system fory without jump incorporated into the dynamical system are given in Figure 3 below.Figure 3
The graph of interaction coefficientsδ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures3(a)–3(i) show the graphs of the ε suboptimal interaction measure of fluctuation coefficient parameters δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.Table3 gives the drift coefficient’s parameter estimates β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k for the dynamical system for p without jump.Table 3
Estimatesm^k, β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, m^k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, m^k,β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
β1m^k,k
γ1,1m^k,k
γ1,2m^k,k
γ1,3m^k,k
β2m^k,k
γ2,1m^k,k
γ2,2m^k,k
γ2,3m^k,k
β3m^k,k
γ3,1m^k,k
γ3,2m^k,k
γ3,3m^k,k
11
0.3462
0.2579
-0.0039
0.0218
0.0029
0.0023
0.0056
0.0124
0
0
0.0432
0.0012
12
0.1681
0.3497
-0.0109
0.0248
0
0
0
0
0.6812
-0.3513
0.0248
0
13
0.1592
0.3755
-0.0102
0.0228
-0.0490
0.0228
0
-0.0027
1.0795
-0.2904
0.0135
0
14
-0.0889
0.5001
-0.0069
0.0257
-0.3700
-0.1689
0
0.0633
-0.6689
-0.1969
0.0166
0.0177
15
0.7025
0.6513
-0.0242
0.0376
-0.0903
-0.1737
0
0.0477
-0.8220
-0.0944
0.0061
0.0291
16
0.6727
0.6513
0.0188
-0.1048
0.3425
-0.1847
0
0.0252
-0.0480
0.0694
-0.0112
0.0165
17
0.3253
0.3674
-0.0103
0.0140
-0.4531
-0.1457
0
0.0638
-3.6240
0.1086
0.0266
0.0569
18
0.1523
0.3433
0.0014
-0.0163
4.0859
0.2320
0.3114
-0.2889
-1.3277
-0.0566
0.0088
0.0353
19
-7.9573
0.3433
-0.2058
1.1496
0.1389
-0.0677
0.0004
0.0065
-0.9232
0.0368
-0.0277
0.0883
20
0.0514
0.3028
0.0041
-0.0192
-0.1464
-0.0409
0.0004
0.0178
-1.9138
0.0530
-0.0104
0.0836
…
…
…
…
…
…
…
…
…
…
…
…
…
495
0.0552
0.2994
0.0015
-0.0063
1.8435
-0.1608
0.1040
-0.0368
0.5815
-0.3597
-0.0050
0.0350
496
0.1799
0.1860
0.0004
-0.0067
1.5682
-0.0268
0.4723
-0.0502
0.8586
-0.2029
-0.0094
0.0241
497
0.8047
0.1923
0.0023
-0.0314
6.6699
-0.2655
0.4723
-0.1649
0.7191
-0.2061
-0.0053
0.0172
498
0.2742
0.2651
0.0020
-0.0145
1.0042
0.0088
0.0226
-0.0315
0.3978
-0.1680
-0.0034
0.0161
499
0.4915
0.2295
-0.0006
-0.0125
1.3074
0.3761
0.0073
-0.0872
0.1425
-0.1899
-0.0026
0.0224
500
0.5659
0.1618
0.0008
-0.0194
0.4040
-0.0889
0.0392
-0.0011
0.3674
-0.2313
-0.0073
0.0331
501
0.4498
0.1679
0.0010
-0.0167
0.4230
-0.1297
0.0434
0.0035
-0.9002
1.1703
-0.0775
0.1011
502
0.4836
0.1850
-0.0001
-0.0139
0.5570
-0.1502
0.0384
0.0022
-0.2313
0.6524
-0.0496
0.0663
503
0.4696
0.1850
0.1224
-0.0919
-0.0441
0.0299
0.0384
-0.0023
3.7804
0.0120
-0.0498
0.0389
504
-0.0456
0.1850
0.0088
-0.0270
0.6112
-0.0820
0.0425
-0.0080
6.4696
0.4005
-0.0950
0.0543
505
0.0464
1.7125
-0.0423
0.1339
0.7135
-0.1115
0.1135
-0.0082
2.2295
0.0897
-0.0357
0.0306
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
0.6765
0.0455
-0.0020
-0.0908
0.2863
-0.2183
0.1891
0.1028
4.3927
3.8144
0.1072
0.0250
1103
1.1804
0.4214
-0.0149
0.0837
-2.1858
0.4491
0.1891
0.1135
-6.1960
0.7446
0.0261
0.0144
1104
0.1069
0.2489
-0.0009
-0.0014
-2.1178
0.3406
0.1959
0.1826
-6.4415
0.0339
-0.0037
0.1429
1105
0.2367
0.0128
0.0065
-0.0019
0.2633
0.0565
0.0742
-0.0997
0.6510
2.4930
0.0714
0.1429
1106
0.1178
0
0
-0.0014
0.1384
0.0784
0.2014
-0.0904
0.6510
2.4930
0.6341
0.1429
1107
0.1466
0.4648
-0.0002
-0.0271
0.5787
-0.0297
0.1305
-0.0979
-4.5897
-0.0971
0.0240
0.0509
1108
0.3240
0.2478
0.1212
-0.0074
0.4293
-0.0678
0.0721
-0.0389
-4.5961
-0.0734
0.0233
0.0507
1109
0.121
0
0
-0.0021
0.2282
-0.0468
0.0721
-0.0133
-4.5961
-2.6776
0.5625
0.4702
1110
0.002
0
0
-0.0056
0.1523
0.0121
0.0011
-0.0129
9.5959
0.9045
-0.1478
0.0499Table3 shows the estimates of the ε suboptimal size m^k, j∈I1,3 and the parameters β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k, for each of the energy commodity data sets. Moreover, d≤r, and the initial real data time is tr=t10.In the following, we give the graph of the drift coefficient’s parameters with estimates in Table3 in Figure 4 below.Figure 4
The graph of interaction coefficientsγ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures4(a)–4(i) show the graphs of the ε suboptimal interaction coefficient parameters γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k. According to (58), the estimate γj,lm^k,k, j≠l, is positive if commodity pl is cooperating with commodity pj, and negative if commodity pl is competing with commodity pj. There is no interaction between the two commodities if γj,lm^k,k=0. It is apparent from the graph of γ1,3m^k,k that coal is competing with natural gas because the estimates of γ1,3m^k,k are mostly negative. Also, it is apparent that natural gas and crude oil are either cooperating or competing, depending on the time period.Figures5(a)–5(c) are the graphs of β1m^k,k, β2m^k,k, and β3m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively.Figure 5
The graph ofβ1m^k,k, β2m^k,k, and β3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)Table4 gives the ε suboptimal estimates of the parameters σ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k for each of the energy commodity data sets.Table 4
Estimatesσ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
σ1,1m^k,k
σ1,2m^k,k
σ1,3m^k,k
σ2,1m^k,k
σ2,2m^k,k
σ2,3m^k,k
σ3,1m^k,k
σ3,2m^k,k
σ3,3m^k,k
11
0
0
0
0
0
0
0
0
0
12
0.0485
0.0004
0.0032
0.2734
0.0166
0
0
0
0.0000
13
0
0
0
0
0
0
0
0
0
14
0.2120
0.1386
0.0133
1.2573
0.4773
0.1195
0
0.0665
0.0086
15
0.4246
0.1318
0.0021
2.1081
0.4894
0.1211
0
0.6107
0.0696
16
0.5538
0.0778
0.1501
0
0.2524
0.0811
0.0651
0.4251
0.0635
17
1.1121
0.0469
0.2230
0
0.1848
0.2463
0
0.4458
0.0478
18
1.5347
0.0180
0.2178
0
0.1877
0.1602
0.5681
0.0592
0.0115
19
1.1315
0.0619
0.2221
0
0.2673
0.2465
0.4999
0.0569
0.0127
20
2.0845
0.0536
0.1866
0
0.1700
0.0781
0.3789
0.3174
0.0046
…
…
…
…
…
…
…
…
…
…
495
0
0.0036
0.0406
0.2286
0.0600
0.0172
0
0.9387
0.0182
496
0.1588
0.0035
0.0107
1.4847
0.3163
0.0102
0
0
0.0016
497
0.1551
0.0009
0.0065
0
0.1453
0
0.7777
0
0.0033
498
0.1576
0.0011
0.0073
0
0.1679
0
0.5334
0
0.0060
499
0.1197
0.0006
0.0059
1.9414
0.2391
0.0172
0.4405
0.1432
0.0097
500
0.3600
0.0001
0.0049
1.9554
0.3960
0.0079
0.6331
0.1410
0.0093
501
0.0514
0.0033
0.0049
2.0436
0.3499
0.0111
0.7690
0.1376
0.0089
502
0.2503
0.0034
0.0042
2.0837
0.1744
0.0132
0.6198
0.1274
0.0066
503
0.1195
0.0147
0.0165
0
0.4283
0.0060
1.1613
0.1530
0.0049
504
0.0974
0.0144
0.0027
0
0.2241
0.0048
0.4778
0.0574
0.0043
505
0.1422
0.0060
0.0131
0
0.2023
0.0054
0.5604
0.0669
0.0004
…
…
…
…
…
…
…
…
…
…
1102
0.1898
0.0016
0.0413
0.8313
0.0767
0.0381
0.6875
0
0.1451
1103
0.2094
0.0015
0.0352
0.8262
0.0673
0.0451
0.7298
0.2808
0.0147
1104
0.1711
0.0011
0.0040
0.6648
0.0915
0.0462
0.5563
0.1831
0.0105
1105
0.1816
0.0012
0.0116
0.6658
0.1049
0.0371
0.6591
0.2874
0.0057
1106
0.1191
0.0011
0.0116
0.6260
0.1155
0.0393
0
0.0196
0.0060
1107
0.0417
0.0012
0.0041
0.4992
0.0781
0.0382
0
0
0.0065
1108
0.1058
0.0033
0.0045
0.0019
0.0589
0.0421
0
0
0.0018
1109
0.1740
0.0021
0
0
0.0446
0.0316
2.1187
0
0.4511
1110
0.2912
0.0021
0.0163
0.0385
0.0342
0.0037
0
1.1563
0.0257Figures6(a)–6(i) are the graphs of σ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ1,1m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively.Figure 6
The graph ofσ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ1,1m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Table5 shows the real and simulated estimates for the spot price processes pjt, j∈I1,n.Table 5
Real and simulated estimates (without jump) for natural gas, crude oil, and coal.
tk
Real
Natural gas
Real
Crude oil
Real
Coal
Simulatedp1sm^k,k
Simulatedp2sm^k,k
Simulatedp3sm^k,k
11
4.0200
4.0500
58.9900
56.5200
16.5900
16.8000
12
3.9900
4.0500
59.5200
59.3099
17.4600
16.8635
13
3.7500
3.6690
61.4500
59.3377
17.8900
17.8086
14
3.7700
3.6341
60.4900
59.4191
17.5500
17.0859
15
3.4100
3.3967
61.1500
59.6974
17.4100
17.0859
16
3.3500
3.3967
62.4800
59.6974
16.7500
17.0859
17
3.4900
3.4537
63.4100
61.2177
17.6600
19.0677
18
3.5500
3.4537
65.0900
61.4561
17.5200
16.0578
19
3.9200
3.8618
66.3100
61.6529
18.5000
19.0677
20
3.8600
3.8618
68.5900
60.9364
19.0600
19.0677
…
…
…
…
…
…
…
495
4.1900
4.0368
107.1800
104.1295
32.7600
31.3108
496
4.3300
4.1868
110.8400
111.1245
33.6500
32.7737
497
4.3300
4.1025
111.7200
112.4675
33.7100
33.4888
498
4.3700
4.0964
111.6800
110.8795
34.7500
35.5907
499
4.3200
4.1042
111.7200
104.2465
34.5400
32.9391
500
4.3500
4.0548
112.3100
109.9535
34.0400
36.2674
501
4.3800
4.0548
112.3800
109.9995
33.1000
36.2674
502
4.5100
4.3249
113.3900
104.3254
33.6700
34.8915
503
4.6000
4.3555
113.0300
113.2356
33.9400
35.0472
504
4.6000
4.3491
110.6000
103.9435
33.8300
32.8992
505
4.5900
4.3609
108.7900
104.9995
32.0200
32.8992
…
…
…
…
…
…
…
1102
3.7200
3.5963
108.2300
110.5149
4.7700
2.8861
1103
3.7300
3.5963
106.2600
105.8076
5.0100
5.6871
1104
3.6800
3.4099
104.7000
105.8076
4.9800
5.3821
1105
3.6600
3.4356
103.6200
105.8076
4.7300
4.9221
1106
3.5900
3.4636
103.2200
106.9547
4.6800
4.2352
1107
3.5200
3.2573
102.6800
105.4047
4.6300
5.8172
1108
3.4900
2.8981
103.1000
102.4928
4.7400
6.0376
1109
3.5100
2.8981
102.8600
102.4928
4.3300
5.1121
1110
3.4800
3.0267
102.3600
102.4928
4.1800
4.8978Figure7 below shows the graph of the real and simulated prices for natural gas, crude oil, and coal data set.Figure 7
Real and simulated prices (without jump) for natural gas, crude oil, and coal.
(a)
(b)
(c)Figures7(a)–7(c) show the graph of the real and simulated spot prices for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively. The red line represents the real data set ptk, while the blue line represent the simulated data set psm^k,k. Here, we begin by using a starting delay of r=10. The simulation starts from tr=t10. It is clear that the graph fits well. To reduce magnitude of error, we increase the magnitude of time delay. We later compare this result with the case where jump is incorporated into the system.


## 7.5.2. Relationship between Natural Gas, Crude Oil, and Coal: With Jump Incorporated

In this subsubsection, we analyze the relationship between natural gas, crude oil, and coal with the jump process. Here, we apply the above conceptual computational algorithm described in this section for the real-time data sets, namely, daily Henry Hub natural gas data set, daily crude oil data set, and daily coal data set for the period of 05/04/2009-01/03/2014, [35–37]. For i∈I1,K∗, K≠0, we use Δti−1=1, ε=0.001, ri−1=10, and di−1=2. The ε-level suboptimal estimates of the parameters at each real data times are described below for each commodity data sets.The parameters corresponding to the model governing natural gas price data set areu1i−1m^k,k, β1i−1m^ki−1,tki−1, κ1,1i−1m^ki−1,tki−1, κ1,2i−1m^ki−1,tki−1, κ1,3i−1m^ki−1,tki−1, γ1,1i−1m^ki−1,tki−1, γ1,2i−1m^ki−1,tki−1, γ1,3i−1m^ki−1,tki−1, δ1,1i−1m^ki−1,tki−1, δ1,2i−1m^ki−1,tki−1, δ1,3i−1m^ki−1,tki−1, σ1,1i−1m^ki−1,tki−1, σ1,2i−1m^ki−1,tki−1, and σ1,3i−1m^ki−1,tki−1. The parameters corresponding to the model governing crude oil price data set are u2i−1m^ki−1,tki−1, β2i−1m^ki−1,tki−1, κ2,1i−1m^ki−1,tki−1, κ2,2i−1m^ki−1,tki−1, κ2,3i−1m^ki−1,tki−1, γ2,1i−1m^ki−1,tki−1, γ2,2i−1m^ki−1,tki−1, γ2,3i−1m^ki−1,tki−1, δ2,1i−1m^ki−1,tki−1, δ2,2i−1m^ki−1,tki−1, δ2,3i−1m^ki−1,tki−1, σ2,1i−1m^ki−1,tki−1, σ2,2i−1m^ki−1,tki−1, and σ2,3i−1m^ki−1,tki−1, while the parameters corresponding to the model governing coal price data set are u3i−1m^ki−1,tki−1, β3i−1m^ki−1,tki−1, κ3,1i−1m^ki−1,tki−1, κ3,2i−1m^ki−1,tki−1, κ3,3i−1m^ki−1,tki−1, γ3,1i−1m^ki−1,tki−1, γ3,2i−1m^ki−1,tki−1, γ3,3i−1m^ki−1,tki−1, δ3,1i−1m^ki−1,tki−1, δ3,2i−1m^ki−1,tki−1, δ3,3i−1m^ki−1,tki−1, σ3,1i−1m^ki−1,tki−1, σ3,2i−1m^ki−1,tki−1, and σ3,3i−1m^ki−1,tki−1.For the sake of simplicity and in order to be able to compare our results in this subsection with the results in Subsection7.5.1, for each j,l∈I1,n, we rewrite the parameters uji−1m^ki−1,tki−1, βji−1m^ki−1,tki−1, κj,li−1m^ki−1,tki−1, γj,li−1m^ki−1,tki−1, δj,li−1m^ki−1,tki−1, and σj,li−1m^ki−1,tki−1 after they have been estimated as ujm^k,k, βjm^k,k, κj,lm^k,k, γj,lm^k,k, δj,lm^k,k, and σj,lm^k,k.We give some results for the jump times of the systemTii∈I1,K∗.Table6 shows some result for the jump times of the system (y,p). These results are derived by recording the times at which an entry of a commodity differ by twice the standard deviation or more from the mean of that commodity. These times are now combined into a single array and sorted out in an increasing order.Table 6
Some result for the jump times of the system (y,p).
T
17
44
61
87
157
200
422
464
483
502
722
754
870
930
1113We give the estimate of the jump coefficient matricesΠi and Θi defined in (34) in Table 7 below.Table 7
Estimatesπ1i, π2i, π3i, θ1i, θ2i, and θ3i.
i
Ti
Natural gas
Crude oil
Coal
Ti
Natural gas
Crude oil
Coal
Ti
Π1i
Π2i
Π3i
Ti
θ1i
θ2i
θ3i
1
17
1.0031
1.1219
1.0256
17
1.0049
1.1219
1.0493
2
44
0.9213
0.9727
1.0410
44
0.9352
1.0084
0.9249
3
61
0.9482
0.9671
0.9661
61
0.9997
0.9427
0.9404
4
87
0.8859
0.9974
0.9653
87
0.7389
1.0452
0.9905
5
157
1.0435
0.9350
1.0432
157
1.0933
1.0019
1.0049
6
200
1.0309
1.0199
1.0382
200
0.9826
1.0210
0.9794
7
422
1.0270
0.9775
0.9669
422
0.9706
0.9939
0.9917
8
464
0.9581
1.0462
1.0523
464
1.0128
1.0508
1.0324
9
483
0.9765
0.9787
1.0291
483
1.0382
1.0328
1.0246
10
502
1.0532
1.0737
1.0136
502
1.0359
1.0073
1.0162
11
722
0.9812
0.9959
0.9919
722
0.9700
0.9695
1.0011
12
754
1.0003
1.0009
0.9189
754
1.0137
0.9987
1.3481
13
870
1.0579
0.9921
1.1378
870
1.0328
1.0033
1.1420
14
930
1.0275
0.9907
0.9978
930
0.9995
0.9812
1.1848
15
1113
1.0009
0.9960
1.0706
1113
0.9304
0.9801
0.9897Table8 gives the estimates u1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for the decoupled dynamical system for y with jump.Table 8
Estimatesm^k, u1m^k,k, κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, m^k, u2m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, m^k, u3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k.
tk
m^k
Natural gas
Crude oil
Coal
u1m^k,k
κ1,1m^k,k
κ1,2m^k,k
κ1,3m^k,k
u2m^k,k
κ2,1m^k,k
κ2,2m^k,k
κ2,3m^k,k
u3m^k,k
κ3,1m^k,k
κ3,2m^k,k
κ3,3m^k,k
×10−16
×10−16
×10−16
×10−16
×10−16
×10−16
11
1
4.1593
0
0
0
57.7000
0
0
0
16.7407
0
0
0
12
4
4.2000
0
0
0
58.6313
0.0011
0.0310
-0.0012
16.2395
0
0
-0.0376
13
6
4.0616
0.0679
-0.0054
-0.0035
58.5378
-0.0035
0.0205
0.0032
16.2680
0
0
0.1069
14
2
4.0616
-0.0242
-0.0179
-0.0035
61.4809
0.0020
0.0098
0
15.5249
0
0
-0.0294
15
7
4.0910
0.6416
-0.2898
0.0078
22.5758
-0.0057
0.0085
0.0012
18.7073
0.0009
-0.0021
0.0318
16
4
4.0160
0
0
0.0078
59.6867
-0.0051
0.0080
0
17.0060
0
-0.0021
0
17
2
4.9575
0
0
0.0078
60.3710
-0.0005
0.0207
0
12.8918
-0.0005
-0.0002
0.0318
18
8
4.9575
-0.1947
0
0.0078
62.3437
0.0005
-0.0008
0
16.5954
0.0002
0.0008
0.0662
19
4
3.3190
-0.4472
0.6760
0.0078
74.6911
-0.0008
-0.0019
0
17.9932
0
-0.0002
0
20
1
3.4762
-0.2540
-0.0048
0.0078
65.9190
0.0026
-0.0006
1
17.6485
0
0
0.0499
…
…
…
…
…
…
…
…
…
…
…
…
…
…
494
1
4.1457
0
0
-0.0001
115.1875
0.0002
0.0053
0
33.3359
0.0003
-0.0003
0.0326
495
1
4.2877
0.1184
-0.0014
0
124.5218
0.0008
0.0056
0
30.1732
0.0002
-0
0.0412
496
1
4.2238
0.2582
0.0011
0.0003
106.8349
0.0003
0.0113
-0.0006
34.9907
0.0034
0
0.0097
497
5
4.0998
0.0477
-0.0006
-0.0002
108.4725
-0.0003
0.0162
-0.0033
33.3388
0.0002
0
0.0443
498
8
4.0592
0.0201
0.0010
0
104.8926
0
0
0.0003
35.1174
0
0.0001
0.0207
499
1
4.3433
0.2118
-0.0014
0
109.2551
-0.0002
0.0048
0.0003
33.2862
0.0010
0.0001
0.0068
500
4
2.4519
0
0
0
111.7067
0
0
0
36.1647
0.0003
0.0003
0.0079
501
1
4.2415
0.4108
0
-0.0015
110.7517
-0.0006
0.0026
0.0009
34.9145
-0.0001
-0.0004
0.0407
502
2
4.3633
0.3210
0.0002
-0.0001
103.6326
-0.0019
0.0023
-0.0002
34.8337
-0.0001
-0.0001
0.0140
503
2
4.2911
0.1276
0.0003
0.0043
112.1547
-0.0033
0.0030
0.0027
35.8389
0.0005
0.0001
0.0211
504
7
4.5942
-0.0125
-0.0002
-0.0031
111.1278
0.0010
0.0072
0.0006
33.6875
-0.0021
0
0.0268
505
2
3.1882
0.0666
0.0009
0
106.1919
-0.0009
0.0110
0.0011
33.6640
-0.0011
-0.0002
0.0231
…
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
1
3.5909
0
0
0.0008
110.3777
0.0006
0.0045
0
5.1761
0.0067
-0.0029
-0.0044
1103
6
3.5303
0.1166
0.0002
-0.0003
111.1585
-0.0003
0.0083
0
5.4558
-0.0019
0.0014
0.0600
1104
4
3.5314
0.0809
0.0018
0
109.0996
-0.0007
0.0095
0.0013
4.8000
0.0005
0.0006
0.1742
1105
1
3.7100
0.2234
-0.0013
-0.0015
106.5667
0.0033
0.0073
-0.0020
5.4226
-0.0082
0.0020
0.0932
1106
8
3.4084
0.1098
0.0001
0
106.5989
0.0003
0.0030
0.0023
5.3360
-0.0023
0.0005
0.0956
1107
7
3.5520
0.1086
0.0001
-0.0070
103.4473
-0.0020
0.0037
-0.0045
4.3586
-0.0005
0.0004
0.1418
1108
5
3.9233
0.0601
0.0007
0
102.8550
0
0.0040
0
4.6582
-0.0010
0
0.1388
1109
8
3.5328
0.0417
0
0
103
-0.0002
0.0089
-0.0005
4.9663
-0.0019
0.0008
0.1279
1110
5
3.8399
0.0212
0.0004
0
102.8800
0
0
0
4.7286
-0.0037
-0.0030
0.0740Figure8 shows the parameter estimates u1m^k,k, u2m^k,k, and u3m^k,k for the decoupled dynamical system for y with jump.Figure 8
The graph of mean levelu1m^k,k, u2m^k,k, and u3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)Figures8(a)–8(c) are the graphs of u1m^k,k, u2m^k,k, and u3m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively. By plotting the real data sets, it is easily seen that the graphs of u1m^k,k, u2m^k,k, and u3m^k,k are similar to the graph of the Henry Hub natural gas, crude oil, and coal data set, respectively. We expect this to happen because uj, j∈I1,3, are the equilibrium spot price processes of (5).The graph of the interaction parametersκ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k for the decoupled dynamical system for y with jump and estimates in Table 8 are given in Figure 9 below.Figure 9
The graph of interaction coefficientsκ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures9(a)–9(i)) show the graph of the ε suboptimal interaction coefficient parameters κ1,1m^k,k, κ1,2m^k,k, κ1,3m^k,k, κ2,1m^k,k, κ2,2m^k,k, κ2,3m^k,k, κ3,1m^k,k, κ3,2m^k,k, and κ3,3m^k,k. The interaction coefficients κj,l, j≠l, are negligible, because each estimate is <<10−15. Thus, this shows that the model describing the mean spot price, yj, is mainly characterized by the market potential κj,ji−1uji−1−yjyj, j∈I1,n, i∈I1,K∗.Table9 below shows the estimates of the diffussion coefficient’s parameters δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k,δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k for the decoupled dynamical system for y with jump.Table 9
Estimatesδ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
δ1,1m^k,k
δ1,2m^k,k
δ1,3m^k,k
δ2,1m^k,k
δ2,2m^k,k
δ2,3m^k,k
δ3,1m^k,k
δ3,2m^k,k
δ3,3m^k,k
11
0.0062
0.0010
0.0001
1.5277
0.0078
0.0011
0
0
0.0218
12
0.0182
0.9002
0
1.6227
0.0010
0
0
0
0.0988
13
0.0239
0.0802
1.7280
0
1.7694
0
0.6374
0.6374
0.0959
14
0
0.0001
0.6027
2.3258
0
0
1.6564
1.6564
0.0847
15
0
0.8001
0.6210
2.3252
0
0
1.6650
1.6650
0.0111
16
0.0455
0.0007
3.6877
2.3217
0
1.2215
1.6724
1.6724
0
17
0
0.9876
0
1.6425
0
0
1.7719
1.7719
0
18
3.0410
0.9351
0
1.3105
0
0.1070
1.7630
1.7630
0.0434
19
2.7713
0.6680
0
1.1052
0
0
1.7400
1.7400
0
20
2.8461
1.7795
0
0.1196
0
0.0983
0
0.4555
0
…
…
…
…
…
…
…
…
…
…
495
1.1229
0
0.0584
0.5488
0.1104
0.0761
0
0
1.3987
496
0.6946
0
0.6613
0.5767
0.0715
0.0610
0
0
1.3017
497
1.1229
0.0095
0.0988
0.6499
0.0870
0.0633
1.1317
1.1317
1.3069
498
0.6946
0.0101
0
0
0
0.0320
1.0294
1.0294
1.5410
499
0.7353
0.0066
0.0384
0
0.0922
0.0330
0.7317
0.7317
1.2225
500
1.7509
0.0069
0.0283
0.4307
0.4545
0.0413
0.4826
0.4826
1.2254
501
2.1299
0.0077
0.0282
0.5043
0.7873
0.0308
0.4272
0.4272
1.5587
502
0.9778
0.0077
0
0.2878
0
0
0.5239
0.5239
1.8713
503
0.9872
0
0
0.2909
0
0
1.4523
1.4523
1.8874
504
1.1329
0
0
0.3707
0.4261
0
0
0
0
505
1.9178
0
0
0.3812
0.7292
0.1724
0
0
0
…
…
…
…
…
…
…
…
…
…
1102
0
0.0331
0.7183
0.9297
0.0434
0.0680
0
0
1.1355
1103
1.5077
0.0626
0.2048
1.1017
0.0421
0.1510
0
0
1.4133
1104
0.4444
0.0435
0.4622
0.1939
0.1078
0
0.0814
0
1.1672
1105
3.5933
0
0.3646
0.1922
0
0.7273
0.2726
0.2726
1.3023
1106
2.4964
0
0.3919
0
0.0684
1.0179
0.3296
0.3296
1.4111
1107
2.4600
0
0.8995
0.2001
0.1510
0.9354
0
0
1.7245
1108
2.0262
0
0.6325
0.3781
0.0814
0.8825
0.1878
0.1878
1.0915
1109
1.7828
0
0.6116
0.4024
0.0332
0.8812
0
0
1.3191
1110
1.2706
0
0.1001
0.3252
0.0155
0.8078
0
0
1.0233The graph of the diffusion coefficient’s parameter for the decoupled dynamical system fory with jump are given in Figure 10 below.Figure 10
The graph of interaction coefficientsδ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures10(a)–10(i) show the graph of the ε- sub-optimal interaction measure of fluctuation coefficient parameters δ1,1m^k,k, δ1,2m^k,k, δ1,3m^k,k, δ2,1m^k,k, δ2,2m^k,k, δ2,3m^k,k, δ3,1m^k,k, δ3,2m^k,k, and δ3,3m^k,k.Table10 gives the drift coefficient’s parameter estimates β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k for the dynamical system for p with jump.Table 10
Estimatesβ1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k.
tK
Natural gas
Crude oil
Coal
β1m^k,k
γ1,1m^k,k
γ1,2m^k,k
γ1,3m^k,k
β2m^k,k
γ2,1m^k,k
γ2,2m^k,k
γ2,3m^k,k
β3m^k,k
γ3,1m^k,k
γ3,2m^k,k
γ3,3m^k,k
11
0
0
0
0
0
0
0
0
0
0
0
0
12
0.1681
0.3497
-0.0109
0.0248
-0.4815
0.1626
-0.4066
-0.0123
0.7665
-0.3259
0.0205
-0.0198
13
0.1592
0.3755
-0.0102
0.0228
-0.7778
-0.5752
-0.0578
0.1870
1.0795
-0.2904
0.0135
-0.0217
14
7.3439
0.3755
0.0488
-0.5478
1.7680
-0.5555
-0.2058
0.0291
0.8543
-0.2056
0.0034
-0.0064
15
0.3336
0.3652
-0.0127
0.0213
-0.8999
0.0601
-0.1110
0.0405
0.5144
-0.1264
-0.0062
0.0127
16
0.4709
0.2780
-0.0116
0.0104
-0.8999
0.0601
-0.1110
-0.0292
0.0017
-0.0002
0
0
17
0.3277
0.2780
0.0768
-0.2633
1.3349
0.0027
-0.0330
-0.0750
-0.6285
-0.0569
0.0016
0.0262
18
0.3277
1.3156
-0.1491
-0.1646
1.5419
-0.0088
0.1205
-0.0892
-1.3275
-0.0703
0.0080
0.0386
19
0
0
0
0
-0.1785
-0.0368
-0.0062
0.0189
-0.8091
-0.0001
-0.0083
0.0466
20
0.6985
0.4990
-0.0069
-0.0187
-0.1513
-0.0778
-0.0096
0.0255
-0.0182
0.0080
0.0001
0.0003
…
…
…
…
…
…
…
…
…
…
…
…
…
495
0.1201
0.2264
0.0007
-0.0056
0.2756
0.1131
-0.9587
-0.0087
-0.0288
-0.0780
-0.0019
0.0136
496
0.1809
0.2085
0.0009
-0.0082
0.2898
-0.0431
0.7329
-0.0133
0.1324
-0.1434
-0.0018
0.0158
497
0.2442
0.1597
0.0007
-0.0093
3.1030
-0.0495
-0.0462
-0.0862
0.9772
-0.2426
-0.0065
0.0177
498
0.2742
0.2651
0.0020
-0.0145
1.3147
0.0148
-0.0009
-0.0411
0.2770
-0.1888
-0.0037
0.0217
499
0.3320
0.3298
-0.0009
-0.0070
0.8430
0.0070
0.0283
-0.0265
0.0931
-0.1551
-0.0041
0.0237
500
0.5035
0.2337
-0.0007
-0.0128
1.3320
0.3949
-0.3308
-0.0838
0.4175
-0.2331
-0.0034
0.0222
501
0.6328
0.2612
-0.0034
-0.0077
0.3251
-0.1148
0.0548
0.0042
0.7896
-0.2546
-0.0093
0.0305
502
0.5403
0.2457
-0.0014
-0.0113
0.4863
-0.1315
0.0343
0.0020
3.7990
0.0420
-0.0524
0.0421
503
0.4794
0.2098
-0.0028
-0.0050
0.1239
0.0013
0.0202
-0.0040
9.6735
0.0736
-0.1152
0.0648
504
-0.3308
-0.5600
0.0258
-0.0737
0.2867
-0.0391
0.0009
-0.0034
4.2547
0.3506
-0.0669
0.0382
505
1.1680
0.8346
-0.0274
0.0542
0.1198
-0.0588
-0.3412
0.0092
2.2295
0.0897
-0.0357
0.0306
…
…
…
…
…
…
…
…
…
…
…
…
…
1102
0.6765
0.0455
-0.0020
-0.0908
0.4026
-0.2544
0.2045
0.1058
-5.9294
0.6777
0.0292
0.0068
1103
1.1804
0.4214
-0.0149
0.0837
-0.6549
0.1780
0.0070
0.0018
-6.3380
0.7440
0.0291
0.0106
1104
0.1069
0.2489
-0.0009
-0.0014
-2.1178
0.3406
0.1959
0.1826
-3.8701
0.5681
0.0157
0.0021
1105
0.0139
0.2777
-0.0001
-0.0008
0.3958
-0.0274
0.0642
-0.0620
-4.0701
0.1880
0.0091
0.0514
1106
-0.2513
0.4043
0.0031
-0.0164
0.4097
0.0060
0.1536
-0.0907
-5.0668
0.3261
0.0178
0.0419
1107
0.0670
0.3163
-0
-0.0145
0.2906
0.0485
0.2310
-0.0989
-5.0668
0.4016
0.0308
0.1474
1108
1.0112
0.6861
-0.0091
-0.0107
0.4281
0.0048
0.1337
-0.0933
-5.0668
0.4650
0.0304
0.1295
1109
0.5020
0.5370
-0.0030
-0.0375
0.3645
-0.0168
0.1078
-0.0641
-5.0668
0.4156
0.0311
0.1396
1110
0.1420
0.3295
0.0009
-0.0484
0.1728
-0.0189
-0.0164
-0.0230
-6.6650
0.4509
0.0099
0.0831Table10 shows the estimates of the ε suboptimal size m^k and the parameters m^k,β1m^k,k, γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, m^k, β2m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, m^k,β3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k, for each of the energy commodity data sets. According to (58), the estimate γj,lm^k,k, j≠l, is positive if commodity pl is cooperating with commodity pj, and negative if commodity pl is competing with commodity pj. There is no interaction between the two commodities if γj,lm^k,k=0. It is apparent from the graph (from γ1,3m^k,k in Column 6) that coal is competing with natural gas during this period because the estimates of γ1,3m^k,k are mostly negative. It is apparent that natural gas and crude oil are either cooperating or competing, depending on the time period.The graph of the drift coefficient’s parameters with estimates in Table10 are given in Figure 11 below.Figure 11
The graph of interaction coefficientsγ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figures11(a)–11(i) show the graph of the ε suboptimal interaction coefficient parameters γ1,1m^k,k, γ1,2m^k,k, γ1,3m^k,k, γ2,1m^k,k, γ2,2m^k,k, γ2,3m^k,k, γ3,1m^k,k, γ3,2m^k,k, and γ3,3m^k,k. According to (58), the estimate γj,lm^k,k, j≠l, is positive if commodity pl is cooperating with commodity pj, and negative if commodity pl is competing with commodity pj. There is no interaction between the two commodities if γj,lm^k,k=0. It is apparent from the graph of γ1,3m^k,k that coal is competing with natural gas because the estimates of γ1,3m^k,k are mostly negative. This shows that for fixed price p1 of natural gas, for every $1 increase in the price of coal, the price of natural gas decreases on the average by 0.0019. Also, it is apparent that natural gas and crude oil are either cooperating or competing, depending on the time period.Figures12(a)–12(c) are the graphs of β1m^k,k, β2m^k,k, and β3m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively.Figure 12
The graph ofβ1m^k,k, β2m^k,k, and β3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)Table11 gives the ε suboptimal estimates of the parameters u1m^k,k, σ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, u2m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, u3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k for each of the energy commodity data sets.Table 11
Estimatesσ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k.
tk
Natural gas
Crude oil
Coal
σ1,1m^k,k
σ1,2m^k,k
σ1,3m^k,k
σ2,1m^k,k
σ2,2m^k,k
σ2,3m^k,k
σ3,1m^k,k
σ3,2m^k,k
σ3,3m^k,k
11
0
0
0
0
0
0
0
0.1303
0
12
0.0485
0.0004
0.0032
0.2734
0.0166
0
0.0513
0
0.0000
13
0.7333
0
0
0.9445
0
0
0.2489
0
0
14
0.2120
0.1386
0.0133
0.3877
0.4773
0.1195
0.1365
0.0665
0.0086
15
0.4246
0.1318
0.0021
0.03341
0.4894
0.1211
0.0112
0.6107
0.0696
16
0.5538
0.0778
0.1501
0.07751
0.2524
0.0811
0.0651
0.4251
0.0635
17
0.3907
0.0469
0.2230
0.08746
0.1848
0.2463
0
0.4458
0.0478
18
0.3523
0.0180
0.2178
0.04291
0.1877
0.1602
0.5681
0.0592
0.0115
19
0.5116
0.0619
0.2221
0.03266
0.2673
0.2465
0.4999
0.0569
0.0127
20
0.6431
0.0536
0.1866
0.0939
0.1700
0.0781
0.3789
0.3174
0.0046
…
…
…
…
…
…
…
…
…
…
495
0
0.0036
0.0406
0.2286
0.0600
0.0172
0.0110
0.9387
0.0182
496
0.1588
0.0035
0.0107
0.08183
0.3163
0.0102
0
0
0.0016
497
0.1551
0.0009
0.0065
0.07869
0.1453
0.4821
0.7777
0
0.0033
498
0.1576
0.0011
0.0073
0.0120
0.1679
0.3786
0.5334
0
0.0060
499
0.1197
0.0006
0.0059
0.0721
0.2391
0.0172
0.4405
0.1432
0.0097
500
0.3600
0.0001
0.0049
0.0273
0.3960
0.0079
0.6331
0.1410
0.0093
0.5010
0.0514
0.0033
0.0049
0.0182
0.3499
0.0111
0.7690
0.1376
0.0089
0.5020
0.2503
0.0034
0.0042
0.0222
0.1744
0.0132
0.6198
0.1274
0.0066
0.5030
0.1195
0.0147
0.0165
0
0.4283
0.0060
0
0.1530
0.0049
0.5040
0.0974
0.0144
0.0027
0
0.2241
0.0048
0.4778
0.0574
0.0043
0.5050
0.1422
0.0060
0.0131
0.0085
0.2023
0.0054
0.5604
0.0669
0.0004
…
…
…
…
…
…
…
…
…
…
1102
0.1898
0.0016
0.0413
0.8313
0.0767
0.0381
0.6875
0
0.1451
1103
0.2094
0.0015
0.0352
0.8262
0.0673
0.0451
0.7298
0.2808
0.0147
1104
0.1711
0.0011
0.0040
0.6648
0.0915
0.0462
0.5563
0.1831
0.0105
1105
0.1816
0.0012
0.0116
0.6658
0.1049
0.0371
0.6591
0.2874
0.0057
1106
0.1191
0.0011
0.0116
0.6260
0.1155
0.0393
0
0.0196
0.0060
1107
0.0417
0.0012
0.0041
0.4992
0.0781
0.0382
0.0271
0.2559
0.0065
1108
0.1058
0.0033
0.0045
0.0019
0.0589
0.0421
0.6209
0.8289
0.0018
1109
0.1740
0.0021
0
0.0305
0.0446
0.0316
0.8431
0.2366
0.4511
1110
0.2912
0.0021
0.0163
0.0385
0.0342
0.0037
0.2910
0.0489
0.0257Figures13(a)–13(c) are the graphs of σ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k against time tk for the daily Henry Hub natural gas price data set [36], daily crude oil price data set [37], and daily coal price data set, respectively.Figure 13
The graph ofσ1,1m^k,k, σ1,2m^k,k, σ1,3m^k,k, σ2,1m^k,k, σ2,2m^k,k, σ2,3m^k,k, σ3,1m^k,k, σ3,2m^k,k, and σ3,3m^k,k for natural gas, crude oil, and coal, respectively.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Table12 shows the real and simulated estimates for the spot price processes pjt, j∈I1,n.Table 12
Real and simulated estimates (with jump) for natural gas, crude oil, and coal.
tk
Natural gas
Crude oil
Coal
Real
Simulatedp1sm^k,k
Real
Simulatedp2sm^k,k
Real
Simulatedp3sm^k,k
11
4.0200
4.0500
58.9900
58.5200
16.5900
16.6000
12
3.9900
3.9600
59.5200
59.5099
17.4600
17.4635
13
3.7500
3.6690
61.4500
60.3377
17.8900
17.8886
14
3.7700
3.7341
60.4900
59.4191
17.5500
17.5188
15
3.4100
3.3967
61.1500
60.1580
17.4100
17.4188
16
3.3500
3.3947
62.4800
62.5028
16.7500
16.7789
17
3.4900
3.3900
63.4100
63.5524
17.6600
17.5341
18
3.5500
3.4957
65.0900
64.9224
17.5200
17.8130
19
3.9200
3.8743
66.3100
66.8239
18.5000
18.8453
20
3.8600
3.8241
68.5900
68.1206
19.0600
18.9453
…
…
…
…
…
…
…
494
4.2300
4.2295
106.7000
106.6374
33.2200
33.1852
495
4.1900
4.2191
107.1800
106.9973
32.7600
32.6677
496
4.3300
4.3414
110.8400
111.0084
33.6500
33.8070
497
4.3300
4.3261
111.7200
111.7084
33.7100
33.7061
498
4.3700
4.3863
111.6800
111.2084
34.7500
35.7057
499
4.3200
4.2167
111.7200
111.6126
34.5400
33.5457
500
4.3500
4.3577
112.3100
112.4358
34.0400
34.2862
501
4.3800
4.3535
112.3800
112.5682
33.1000
33.1330
502
4.5100
4.4389
113.3900
113.2925
33.6700
33.6216
503
4.6000
4.6139
113.0300
113.3077
33.9400
33.9216
504
4.6000
4.5964
110.6000
110.8350
33.8300
33.9216
505
4.5900
4.5564
108.7900
108.7598
32.0200
31.9867
…
…
…
…
…
…
…
1102
3.7200
3.6616
108.2300
108.2354
4.7700
4.5482
1103
3.7300
3.6477
106.2600
106.1451
5.0100
5.1120
1104
3.6800
3.6748
104.7000
104.6723
4.9800
5.1180
1105
3.6600
3.6861
103.6200
104.6723
4.7300
4.7893
1106
3.5900
3.6436
103.2200
102.9765
4.6800
4.7074
1107
3.5200
3.5213
102.6800
102.7652
4.6300
4.6419
1108
3.4900
3.4564
103.1000
102.9765
4.7400
4.4016
1109
3.5100
3.2596
102.8600
102.9652
4.3300
4.1826
1110
3.4800
3.4604
102.3600
102.4345
4.1800
4.4606Figure14 shows the graph of the real and simulated prices (with jump) for natural gas, crude oil, and coal data set.Figure 14
Real and simulated prices (with jump) for natural gas, crude oil, and coal.
(a)
(b)
(c)Figures14(a)–14(c) show the graph of the real and simulated spot prices for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively. The red line represents the real data set p, while the blue line represent the simulated data set pm^k,ks. It is clear that the graph fits well. To reduce magnitude of error, we increase the magnitude of time delay. It is obvious that these curves fit better than the curves in Figure 7. It follows that the interconnected dynamical system with jump process incorporated into it performs better than the one without jump.


## 8. Forecasting

In this section, we shall sketch an outline about forecasting problem for the case where there is no jump. The sketch for the case where jump exist is similar. Anε suboptimal simulated value (ysm^ki−1,tki−1, psm^ki−1,tki−1) at time tki−1, i∈I1,K∗, is used to define a forecast (yfm^ki−1,tki−1, pfm^ki−1,tki−1) for (ytki−1,ptki−1) at the time tki−1 for the system of energy commodity model.


### 8.1. Forecasting for Energy Commodity Model

In the context of Section6.1, for i∈I1,K∗, we begin forecasting from time tki−1. Using the data set up to time tk−1i−1, we compute m^ai−1, m^ai−1, ujm^ai−1,tai−1, βjm^ai−1,tai−1, κj,lm^ai−1,tai−1, γj,lm^ai−1,tai−1, δj,lm^ai−1,tai−1, σj,lm^ai−1,tai−1, and j,l∈I1,3 for a∈I0,k−1. We assume that we have no information about the real data set yjtai−1a=kNi−1. Under these considerations, imitating the computational procedure outlined in Section 7 and using solutions to (66) and (67), we find the estimate of the forecast yfm^ki−1,tki−1 and pfm^ki−1,tki−1 at time tki−1 as follows:
(87)yjfm^ki−1,tki−1=yjsm^k−1i−1,tk−1i−1+ujm^k−1i−1,tk−1i−1−yjsm^k−1i−1,tk−1i−1κj,jm^k−1i−1,tk−1i−1yjsm^k−1i−1,tk−1i−1+∑l≠jnκj,lylsm^k−1i−1,tk−1i−1Δt+δj,jm^k−1i−1,tk−1i−1ujm^k−1i−1,tk−1i−1−yjsm^k−1i−1,tk−1i−1Wj,jk+ujm^k−1i−1,tk−1i−1−yjsm^k−1i−1,tk−1i−1∑l≠jnδj,lylsm^k−1i−1,tk−1i−1Wj,lk,pjfm^ki−1,tki−1=pjsm^k−1i−1,tk−1i−1+pjsm^k−1i−1,tk−1i−1γj,jm^k−1i−1,tk−1i−1yjsm^k−1i−1,tk−1i−1−pjsm^k−1i−1,tk−1i−1+βjm^k−1i−1,tk−1i−1+∑l≠jnγj,lm^k−1i−1,tk−1i−1plsm^k−1i−1,tk−1i−1Δt+σj,jm^k−1i−1,tk−1i−1pjsm^k−1i−1,tk−1i−1Zj,jk+pjsm^k−1i−1,tk−1i−1∑l≠jnσj,lm^k−1i−1,tk−1i−1plsm^k−1i−1,tk−1i−1Zj,lk,tki−1∈Ti−1,Ti.To find the forecasted valueyjfTi− and pjfTi−, we need to re-estimate π^ji and θ^ji by first simulating
(88)limt→Ti−yjft,Ti−1,yi−1,s≡yjfmNi−1i−1,tNi−1i−1,limt→Ti−pjft,Ti−1,yi−1,s,pi−1,s≡pjfmNi−1i−1,tNi−1i−1,as follows:
(89)yjfmNi−1i−1,tNi−1i−1=yjsmNi−1−1i−1,tNi−1−1i−1+uji−1mNi−1−1i−1,tNi−1−1i−1−yjsmNi−1−1i−1,tNi−1−1i−1∑l=1nκj,li−1mNi−1−1i−1,tNi−1−1i−1ylsmNi−1−1i−1,tNi−1−1i−1Δt+uji−1mNi−1−1i−1,tNi−1−1i−1−yjsmNi−1−1i−1,tNi−1−1i−1δj,ji−1mNi−1−1i−1,tNi−1−1i−1ΔWj,jmNi−1i−1,tNi−1i−1+∑l≠jnδj,li−1mNi−1−1i−1,tNi−1−1i−1ylsmNi−1−1i−1,tNi−1−1i−1ΔWj,lmNi−1i−1,tNi−1i−1,pjfmNi−1i−1,tNi−1i−1=pjsγj,ji−1mNi−1−1i−1,tNi−1−1i−1yjsmNi−1−1i−1,tNi−1−1i−1−pjsmNi−1−1i−1,tNi−1−1i−1+βji−1mNi−1−1i−1,tNi−1−1i−1+∑l≠jnγj,li−1mNi−1−1i−1,tNi−1−1i−1plsmNi−1−1i−1,tNi−1−1i−1Δt+σj,ji−1mNi−1−1i−1,tNi−1−1i−1pjsmNi−1−1i−1,tNi−1−1i−1ΔZj,jmNi−1i−1,tNi−1i−1+pjsmNi−1−1i−1,tNi−1−1i−1∑l≠jnσj,li−1mNi−1−1i−1,tNi−1−1i−1plsmNi−1−1i−1,tNi−1−1i−1ΔZj,lmNi−1i−1,tNi−1i−1,tki−1=tNi−1i−1.Imitating argument in (84), we define
(90)π^ji=yjsmNi−1i−1,tNi−1i−1yjfmNi−1i−1,tNi−1i−1,θ^ji=pjsmNi−1i−1,tNi−1i−1pjfmNi−1i−1,tNi−1i−1.Thus,yjfTi−=π^jiyjfTi−,Ti−1,yi−1,s and pjfTi−=θ^jipjfTi−,Ti−1,yi−1,f,pi−1,f. where the estimates ujm^k−1i−1,tk−1i−1, βjm^k−1i−1,tk−1i−1, κj,lm^k−1i−1,tk−1i−1, γj,lm^k−1i−1,tk−1i−1, δj,lm^k−1i−1,tk−1i−1, and σj,lm^k−1i−1,tk−1i−1, j,l∈I1,3 are estimated with respect to the known past data set up to the time tk−1i−1. We note that ym^ki−1,tki−1f is the ε suboptimal estimate for yjtki−1 at time tki−1.To determine (yfm^k+1i−1,tk+1i−1,pfm^k+1i−1,tk+1i−1), we need ujm^ki−1,tki−1, βjm^ki−1,tki−1, κj,lm^ki−1,tki−1, γj,lm^ki−1,tki−1, δj,lm^ki−1,tki−1, and σj,lm^ki−1,tki−1, j,l∈I1,3. Since we only have information of real data up to time tk−1, we use the forecasted estimate yjfm^ki−1,tki−1 as the estimate of yjtki−1 at time tki−1 and to estimate ujm^ki−1,tki−1, βjm^ki−1,tki−1, κj,lm^ki−1,tki−1, γj,lm^ki−1,tki−1, δj,lm^ki−1,tki−1, and σj,lm^ki−1,tki−1, j,l∈I1,3. Hence, we can write ujm^ki−1,tki−1 as
(91)ujm^ki−1,tki−1≡ujm^ki−1,yjtk−m^ki−1+1i−1,yjtk−m^ki−1+2i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,κj,lm^ki−1,tki−1≡κj,lm^ki−1,yjtk−m^ki−1+1i−1,yjtk−m^ki−1+2i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,δj,lm^ki−1,tki−1≡δj,lm^ki−1,yjtk−m^ki−1+1i−1,yjtk−m^ki−1+2i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,βj,lm^ki−1,tki−1≡βj,lm^ki−1,pjtk−m^ki−1+1i−1,pjtk−m^ki−1+2i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,γj,lm^ki−1,tki−1≡γj,lm^ki−1,pjtk−m^ki−1+1i−1,pjtk−m^ki−1+2i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,σj,lm^ki−1,tki−1≡σj,lm^ki−1,pjtk−m^ki−1+1i−1,pjtk−m^ki−1+2i−1,⋯,pjt,k−1i−1,pjfm^ki−1,tki−1,j,l∈I1,n.Forj,l∈I1,n, to find (yjfm^k+2i−1,tk+2i−1, pjfm^k+2i−1,tk+2i−1), we use the estimates
(92)ujm^k+1i−1,tk+1i−1≡ujm^k+1i−1,yjtk−m^ki−1+2i−1,yjtk−m^ki−1+3i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,yjfm^k+1i−1,tk+1i−1,κj,lm^k+1i−1,tk+1i−1≡κj,lm^k+1i−1,yjtk−m^ki−1+2i−1,yjtk−m^ki−1+3i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,yjfm^k+1i−1,tk+1i−1,δj,lm^k+1i−1,tk+1i−1≡δj,lm^k+1i−1,yjtk−m^ki−1+2i−1,yjtk−m^ki−1+3i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,yjfm^k+1i−1,tk+1i−1,βj,lm^k+1i−1,tk+1i−1≡βj,lm^k+1i−1,pjtk−m^ki−1+2i−1,pjtk−m^ki−1+3i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,pjfm^k+1i−1,tk+1i−1,γj,lm^k+1i−1,tk+1i−1≡γj,lm^k+1i−1,pjtk−m^ki−1+2i−1,pjtk−m^ki−1+3i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,pjfm^k+1i−1,tk+1i−1,σj,lm^k+1i−1,tk+1i−1≡σj,lm^k+1i−1,pjtk−m^ki−1+2i−1,pjtk−m^ki−1+3i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,pjfm^k+1i−1,tk+1i−1.Continuing this process in this manner, to findyjfm^k+li−1,tk+li−1,pjfm^k+li−1,tk+li−1, we use the estimates
(93)ujm^k+l−1i−1,tk+l−1i−1≡ujm^k+l−1i−1,yjtk−m^ki−1+li−1,yjtk−m^ki−1+l+1i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,⋯,yjfm^k+l−1i−1,tk+l−1i−1,κj,lm^k+l−1i−1,tk+l−1i−1≡κj,lm^k+l−1i−1,yjtk−m^ki−1+li−1,yjtk−m^ki−1+l+1i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,⋯,yjfm^k+l−1i−1,tk+l−1i−1,δj,lm^k+l−1i−1,tk+l−1i−1≡δj,lm^k+l−1i−1,yjtk−m^ki−1+li−1,yjtk−m^ki−1+l+1i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,⋯,yjfm^k+l−1i−1,tk+l−1i−1,βj,lm^k+l−1i−1,tk+l−1i−1≡βj,lm^k+l−1i−1,pjtk−m^ki−1+li−1,pjtk−m^ki−1+l+1i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,⋯,pjfm^k+l−1i−1,tk+l−1i−1,γj,lm^k+l−1i−1,tk+l−1i−1≡γj,lm^k+l−1i−1,pjtk−m^ki−1+li−1,pjtk−m^ki−1+l+1i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,⋯,pjfm^k+l−1i−1,tk+l−1i−1,σj,lm^k+l−1i−1,tk+l−1i−1≡σj,lm^k+l−1i−1,pjtk−m^ki−1+ii−1,pjtk−m^ki−1+l+1i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,⋯,pjfm^k+l−1i−1,tk+l−1i−1.


#### 8.1.1. Prediction/Confidence Interval for Energy Commodities

In order to be able to assess the future certainty, we also discuss about the prediction/confidence interval. We define the1001−α% confidence interval for the forecast of the state (ym^li−1,tli−1f,pm^li−1,tli−1f) at time tli−1, l≥k, j∈I1,n as
(94)yjfm^li−1,tli−1±z1−α/2sm∧l−1i−1,tl−1i−1j,jyjf1/2,pjfm^li−1,tli−1±z1−α/2sm∧l−1i−1,tl−1i−1j,jpjf1/2,where sj,jm∧l−1i−1,tl−1i−11/2 is the estimate for the sample standard deviation for the forecasted state yj, and sj,jm∧l−1i−1,tl−1i−11/2 is the estimate for the sample standard deviation for the forecasted state pj derived from the following iterative process
(95)yjfm^li−1,tli−1=yjfm^l−1i−1,tl−1i−1+ujm^l−1i−1,tl−1i−1−yjfm^l−1i−1,tl−1i−1κj,jm^l−1i−1,tl−1i−1yjfm^l−1i−1,tl−1i−1+∑l≠jnκj,lylfm^l−1i−1,tl−1i−1Δt+δj,jm^l−1i−1,tl−1i−1ujm^l−1i−1,tl−1i−1−yjfm^l−1i−1,tl−1i−1Wj,jl+ujm^l−1i−1,tl−1i−1−yjfm^l−1i−1,tl−1i−1∑l≠jnδj,lylfm^l−1i−1,tl−1i−1Wj,ll,pjfm^li−1,tli−1=pjfm^l−1i−1,tl−1i−1+pjfm^l−1i−1,tl−1i−1γj,jm^l−1i−1,tl−1i−1yjfm^l−1i−1,tl−1i−1−pjfm^l−1i−1,tl−1i−1+βjm^l−1i−1,tl−1i−1+∑l≠jnγj,lm^l−1i−1,tl−1i−1plfm^l−1i−1,tl−1i−1Δt+σj,jm^l−1i−1,tl−1i−1pjfm^l−1i−1,tl−1i−1Zj,jl+pjfm^l−1i−1,tl−1i−1∑l≠jnσj,lm^l−1i−1,tl−1i−1plfm^l−1i−1,tl−1i−1Zj,ll.It is clear that the 95% confidence interval for the forecast at timetk is
(96)yjfm^ki−1,tki−1−1.96sm∧k−1i−1,k−1j,jyjf1/2,yjfm^ki−1,tki−1+1.96sm∧k−1i−1,k−1j,jyjf1/2,pjfm^ki−1,tki−1−1.96sm∧k−1i−1,k−1j,jpjf1/2,pjfm^ki−1,tki−1+1.96sm∧k−1i−1,k−1j,jpjf1/2,where the lower ends denote the lower bounds of the state estimate and the upper ends denote the upper bounds of the state estimate.


#### 8.1.2. Illustration: Prediction/Confidence Interval for Energy Commodities with No Jump

For the case of no jump, the following graphs show the simulated, forecasted, and 95 percent confidence limit for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively.Figures15(a)–15(c) show the graph of the forecast and 95 percent confidence limit for the case where there is no jump for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively. The figure shows two regions: the simulation region S and the forecast region F. For the simulation region S, we plot the real data set together with the simulated data set as described in Figure 14. For forecast region F, we plot the estimate of the forecast as explained in Section 8.1.1.Figure 15
Real, simulated, forecasted prices and 95% C.I. with no jump.
(a)
(b)
(c)


#### 8.1.3. Illustration: Prediction/Confidence Interval for Energy Commodities with Jump

For the jump process, the following graphs show the forecast and 95 percent confidence limit for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively.Figures16(a)–16(c) show the graphs of the forecast and 95 percent confidence limit for the case where there is jump for the daily Henry Hub Natural gas data set [36], daily Crude Oil data set [37], and daily Coal data set [35], respectively. Figures 16(a)–16(c) show two regions: the simulation region S and the forecast region F. For the simulation region S, we plot the real data set together with the simulated data set as described in Figure 14. For the forecast region F, we plot the estimate of the forecast as explained in Section 8.1.1.Figure 16
Real, simulated, forecasted prices and 95% CI with jump.
(a)
(b)
(c)


## 8.1. Forecasting for Energy Commodity Model

In the context of Section6.1, for i∈I1,K∗, we begin forecasting from time tki−1. Using the data set up to time tk−1i−1, we compute m^ai−1, m^ai−1, ujm^ai−1,tai−1, βjm^ai−1,tai−1, κj,lm^ai−1,tai−1, γj,lm^ai−1,tai−1, δj,lm^ai−1,tai−1, σj,lm^ai−1,tai−1, and j,l∈I1,3 for a∈I0,k−1. We assume that we have no information about the real data set yjtai−1a=kNi−1. Under these considerations, imitating the computational procedure outlined in Section 7 and using solutions to (66) and (67), we find the estimate of the forecast yfm^ki−1,tki−1 and pfm^ki−1,tki−1 at time tki−1 as follows:
(87)yjfm^ki−1,tki−1=yjsm^k−1i−1,tk−1i−1+ujm^k−1i−1,tk−1i−1−yjsm^k−1i−1,tk−1i−1κj,jm^k−1i−1,tk−1i−1yjsm^k−1i−1,tk−1i−1+∑l≠jnκj,lylsm^k−1i−1,tk−1i−1Δt+δj,jm^k−1i−1,tk−1i−1ujm^k−1i−1,tk−1i−1−yjsm^k−1i−1,tk−1i−1Wj,jk+ujm^k−1i−1,tk−1i−1−yjsm^k−1i−1,tk−1i−1∑l≠jnδj,lylsm^k−1i−1,tk−1i−1Wj,lk,pjfm^ki−1,tki−1=pjsm^k−1i−1,tk−1i−1+pjsm^k−1i−1,tk−1i−1γj,jm^k−1i−1,tk−1i−1yjsm^k−1i−1,tk−1i−1−pjsm^k−1i−1,tk−1i−1+βjm^k−1i−1,tk−1i−1+∑l≠jnγj,lm^k−1i−1,tk−1i−1plsm^k−1i−1,tk−1i−1Δt+σj,jm^k−1i−1,tk−1i−1pjsm^k−1i−1,tk−1i−1Zj,jk+pjsm^k−1i−1,tk−1i−1∑l≠jnσj,lm^k−1i−1,tk−1i−1plsm^k−1i−1,tk−1i−1Zj,lk,tki−1∈Ti−1,Ti.To find the forecasted valueyjfTi− and pjfTi−, we need to re-estimate π^ji and θ^ji by first simulating
(88)limt→Ti−yjft,Ti−1,yi−1,s≡yjfmNi−1i−1,tNi−1i−1,limt→Ti−pjft,Ti−1,yi−1,s,pi−1,s≡pjfmNi−1i−1,tNi−1i−1,as follows:
(89)yjfmNi−1i−1,tNi−1i−1=yjsmNi−1−1i−1,tNi−1−1i−1+uji−1mNi−1−1i−1,tNi−1−1i−1−yjsmNi−1−1i−1,tNi−1−1i−1∑l=1nκj,li−1mNi−1−1i−1,tNi−1−1i−1ylsmNi−1−1i−1,tNi−1−1i−1Δt+uji−1mNi−1−1i−1,tNi−1−1i−1−yjsmNi−1−1i−1,tNi−1−1i−1δj,ji−1mNi−1−1i−1,tNi−1−1i−1ΔWj,jmNi−1i−1,tNi−1i−1+∑l≠jnδj,li−1mNi−1−1i−1,tNi−1−1i−1ylsmNi−1−1i−1,tNi−1−1i−1ΔWj,lmNi−1i−1,tNi−1i−1,pjfmNi−1i−1,tNi−1i−1=pjsγj,ji−1mNi−1−1i−1,tNi−1−1i−1yjsmNi−1−1i−1,tNi−1−1i−1−pjsmNi−1−1i−1,tNi−1−1i−1+βji−1mNi−1−1i−1,tNi−1−1i−1+∑l≠jnγj,li−1mNi−1−1i−1,tNi−1−1i−1plsmNi−1−1i−1,tNi−1−1i−1Δt+σj,ji−1mNi−1−1i−1,tNi−1−1i−1pjsmNi−1−1i−1,tNi−1−1i−1ΔZj,jmNi−1i−1,tNi−1i−1+pjsmNi−1−1i−1,tNi−1−1i−1∑l≠jnσj,li−1mNi−1−1i−1,tNi−1−1i−1plsmNi−1−1i−1,tNi−1−1i−1ΔZj,lmNi−1i−1,tNi−1i−1,tki−1=tNi−1i−1.Imitating argument in (84), we define
(90)π^ji=yjsmNi−1i−1,tNi−1i−1yjfmNi−1i−1,tNi−1i−1,θ^ji=pjsmNi−1i−1,tNi−1i−1pjfmNi−1i−1,tNi−1i−1.Thus,yjfTi−=π^jiyjfTi−,Ti−1,yi−1,s and pjfTi−=θ^jipjfTi−,Ti−1,yi−1,f,pi−1,f. where the estimates ujm^k−1i−1,tk−1i−1, βjm^k−1i−1,tk−1i−1, κj,lm^k−1i−1,tk−1i−1, γj,lm^k−1i−1,tk−1i−1, δj,lm^k−1i−1,tk−1i−1, and σj,lm^k−1i−1,tk−1i−1, j,l∈I1,3 are estimated with respect to the known past data set up to the time tk−1i−1. We note that ym^ki−1,tki−1f is the ε suboptimal estimate for yjtki−1 at time tki−1.To determine (yfm^k+1i−1,tk+1i−1,pfm^k+1i−1,tk+1i−1), we need ujm^ki−1,tki−1, βjm^ki−1,tki−1, κj,lm^ki−1,tki−1, γj,lm^ki−1,tki−1, δj,lm^ki−1,tki−1, and σj,lm^ki−1,tki−1, j,l∈I1,3. Since we only have information of real data up to time tk−1, we use the forecasted estimate yjfm^ki−1,tki−1 as the estimate of yjtki−1 at time tki−1 and to estimate ujm^ki−1,tki−1, βjm^ki−1,tki−1, κj,lm^ki−1,tki−1, γj,lm^ki−1,tki−1, δj,lm^ki−1,tki−1, and σj,lm^ki−1,tki−1, j,l∈I1,3. Hence, we can write ujm^ki−1,tki−1 as
(91)ujm^ki−1,tki−1≡ujm^ki−1,yjtk−m^ki−1+1i−1,yjtk−m^ki−1+2i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,κj,lm^ki−1,tki−1≡κj,lm^ki−1,yjtk−m^ki−1+1i−1,yjtk−m^ki−1+2i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,δj,lm^ki−1,tki−1≡δj,lm^ki−1,yjtk−m^ki−1+1i−1,yjtk−m^ki−1+2i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,βj,lm^ki−1,tki−1≡βj,lm^ki−1,pjtk−m^ki−1+1i−1,pjtk−m^ki−1+2i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,γj,lm^ki−1,tki−1≡γj,lm^ki−1,pjtk−m^ki−1+1i−1,pjtk−m^ki−1+2i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,σj,lm^ki−1,tki−1≡σj,lm^ki−1,pjtk−m^ki−1+1i−1,pjtk−m^ki−1+2i−1,⋯,pjt,k−1i−1,pjfm^ki−1,tki−1,j,l∈I1,n.Forj,l∈I1,n, to find (yjfm^k+2i−1,tk+2i−1, pjfm^k+2i−1,tk+2i−1), we use the estimates
(92)ujm^k+1i−1,tk+1i−1≡ujm^k+1i−1,yjtk−m^ki−1+2i−1,yjtk−m^ki−1+3i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,yjfm^k+1i−1,tk+1i−1,κj,lm^k+1i−1,tk+1i−1≡κj,lm^k+1i−1,yjtk−m^ki−1+2i−1,yjtk−m^ki−1+3i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,yjfm^k+1i−1,tk+1i−1,δj,lm^k+1i−1,tk+1i−1≡δj,lm^k+1i−1,yjtk−m^ki−1+2i−1,yjtk−m^ki−1+3i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,yjfm^k+1i−1,tk+1i−1,βj,lm^k+1i−1,tk+1i−1≡βj,lm^k+1i−1,pjtk−m^ki−1+2i−1,pjtk−m^ki−1+3i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,pjfm^k+1i−1,tk+1i−1,γj,lm^k+1i−1,tk+1i−1≡γj,lm^k+1i−1,pjtk−m^ki−1+2i−1,pjtk−m^ki−1+3i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,pjfm^k+1i−1,tk+1i−1,σj,lm^k+1i−1,tk+1i−1≡σj,lm^k+1i−1,pjtk−m^ki−1+2i−1,pjtk−m^ki−1+3i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,pjfm^k+1i−1,tk+1i−1.Continuing this process in this manner, to findyjfm^k+li−1,tk+li−1,pjfm^k+li−1,tk+li−1, we use the estimates
(93)ujm^k+l−1i−1,tk+l−1i−1≡ujm^k+l−1i−1,yjtk−m^ki−1+li−1,yjtk−m^ki−1+l+1i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,⋯,yjfm^k+l−1i−1,tk+l−1i−1,κj,lm^k+l−1i−1,tk+l−1i−1≡κj,lm^k+l−1i−1,yjtk−m^ki−1+li−1,yjtk−m^ki−1+l+1i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,⋯,yjfm^k+l−1i−1,tk+l−1i−1,δj,lm^k+l−1i−1,tk+l−1i−1≡δj,lm^k+l−1i−1,yjtk−m^ki−1+li−1,yjtk−m^ki−1+l+1i−1,⋯,yjtk−1i−1,yjfm^ki−1,tki−1,⋯,yjfm^k+l−1i−1,tk+l−1i−1,βj,lm^k+l−1i−1,tk+l−1i−1≡βj,lm^k+l−1i−1,pjtk−m^ki−1+li−1,pjtk−m^ki−1+l+1i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,⋯,pjfm^k+l−1i−1,tk+l−1i−1,γj,lm^k+l−1i−1,tk+l−1i−1≡γj,lm^k+l−1i−1,pjtk−m^ki−1+li−1,pjtk−m^ki−1+l+1i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,⋯,pjfm^k+l−1i−1,tk+l−1i−1,σj,lm^k+l−1i−1,tk+l−1i−1≡σj,lm^k+l−1i−1,pjtk−m^ki−1+ii−1,pjtk−m^ki−1+l+1i−1,⋯,pjtk−1i−1,pjfm^ki−1,tki−1,⋯,pjfm^k+l−1i−1,tk+l−1i−1.


### 8.1.1. Prediction/Confidence Interval for Energy Commodities

In order to be able to assess the future certainty, we also discuss about the prediction/confidence interval. We define the1001−α% confidence interval for the forecast of the state (ym^li−1,tli−1f,pm^li−1,tli−1f) at time tli−1, l≥k, j∈I1,n as
(94)yjfm^li−1,tli−1±z1−α/2sm∧l−1i−1,tl−1i−1j,jyjf1/2,pjfm^li−1,tli−1±z1−α/2sm∧l−1i−1,tl−1i−1j,jpjf1/2,where sj,jm∧l−1i−1,tl−1i−11/2 is the estimate for the sample standard deviation for the forecasted state yj, and sj,jm∧l−1i−1,tl−1i−11/2 is the estimate for the sample standard deviation for the forecasted state pj derived from the following iterative process
(95)yjfm^li−1,tli−1=yjfm^l−1i−1,tl−1i−1+ujm^l−1i−1,tl−1i−1−yjfm^l−1i−1,tl−1i−1κj,jm^l−1i−1,tl−1i−1yjfm^l−1i−1,tl−1i−1+∑l≠jnκj,lylfm^l−1i−1,tl−1i−1Δt+δj,jm^l−1i−1,tl−1i−1ujm^l−1i−1,tl−1i−1−yjfm^l−1i−1,tl−1i−1Wj,jl+ujm^l−1i−1,tl−1i−1−yjfm^l−1i−1,tl−1i−1∑l≠jnδj,lylfm^l−1i−1,tl−1i−1Wj,ll,pjfm^li−1,tli−1=pjfm^l−1i−1,tl−1i−1+pjfm^l−1i−1,tl−1i−1γj,jm^l−1i−1,tl−1i−1yjfm^l−1i−1,tl−1i−1−pjfm^l−1i−1,tl−1i−1+βjm^l−1i−1,tl−1i−1+∑l≠jnγj,lm^l−1i−1,tl−1i−1plfm^l−1i−1,tl−1i−1Δt+σj,jm^l−1i−1,tl−1i−1pjfm^l−1i−1,tl−1i−1Zj,jl+pjfm^l−1i−1,tl−1i−1∑l≠jnσj,lm^l−1i−1,tl−1i−1plfm^l−1i−1,tl−1i−1Zj,ll.It is clear that the 95% confidence interval for the forecast at timetk is
(96)yjfm^ki−1,tki−1−1.96sm∧k−1i−1,k−1j,jyjf1/2,yjfm^ki−1,tki−1+1.96sm∧k−1i−1,k−1j,jyjf1/2,pjfm^ki−1,tki−1−1.96sm∧k−1i−1,k−1j,jpjf1/2,pjfm^ki−1,tki−1+1.96sm∧k−1i−1,k−1j,jpjf1/2,where the lower ends denote the lower bounds of the state estimate and the upper ends denote the upper bounds of the state estimate.


### 8.1.2. Illustration: Prediction/Confidence Interval for Energy Commodities with No Jump

For the case of no jump, the following graphs show the simulated, forecasted, and 95 percent confidence limit for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively.Figures15(a)–15(c) show the graph of the forecast and 95 percent confidence limit for the case where there is no jump for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively. The figure shows two regions: the simulation region S and the forecast region F. For the simulation region S, we plot the real data set together with the simulated data set as described in Figure 14. For forecast region F, we plot the estimate of the forecast as explained in Section 8.1.1.Figure 15
Real, simulated, forecasted prices and 95% C.I. with no jump.
(a)
(b)
(c)


### 8.1.3. Illustration: Prediction/Confidence Interval for Energy Commodities with Jump

For the jump process, the following graphs show the forecast and 95 percent confidence limit for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively.Figures16(a)–16(c) show the graphs of the forecast and 95 percent confidence limit for the case where there is jump for the daily Henry Hub Natural gas data set [36], daily Crude Oil data set [37], and daily Coal data set [35], respectively. Figures 16(a)–16(c) show two regions: the simulation region S and the forecast region F. For the simulation region S, we plot the real data set together with the simulated data set as described in Figure 14. For the forecast region F, we plot the estimate of the forecast as explained in Section 8.1.1.Figure 16
Real, simulated, forecasted prices and 95% CI with jump.
(a)
(b)
(c)


## 8.1.1. Prediction/Confidence Interval for Energy Commodities

In order to be able to assess the future certainty, we also discuss about the prediction/confidence interval. We define the1001−α% confidence interval for the forecast of the state (ym^li−1,tli−1f,pm^li−1,tli−1f) at time tli−1, l≥k, j∈I1,n as
(94)yjfm^li−1,tli−1±z1−α/2sm∧l−1i−1,tl−1i−1j,jyjf1/2,pjfm^li−1,tli−1±z1−α/2sm∧l−1i−1,tl−1i−1j,jpjf1/2,where sj,jm∧l−1i−1,tl−1i−11/2 is the estimate for the sample standard deviation for the forecasted state yj, and sj,jm∧l−1i−1,tl−1i−11/2 is the estimate for the sample standard deviation for the forecasted state pj derived from the following iterative process
(95)yjfm^li−1,tli−1=yjfm^l−1i−1,tl−1i−1+ujm^l−1i−1,tl−1i−1−yjfm^l−1i−1,tl−1i−1κj,jm^l−1i−1,tl−1i−1yjfm^l−1i−1,tl−1i−1+∑l≠jnκj,lylfm^l−1i−1,tl−1i−1Δt+δj,jm^l−1i−1,tl−1i−1ujm^l−1i−1,tl−1i−1−yjfm^l−1i−1,tl−1i−1Wj,jl+ujm^l−1i−1,tl−1i−1−yjfm^l−1i−1,tl−1i−1∑l≠jnδj,lylfm^l−1i−1,tl−1i−1Wj,ll,pjfm^li−1,tli−1=pjfm^l−1i−1,tl−1i−1+pjfm^l−1i−1,tl−1i−1γj,jm^l−1i−1,tl−1i−1yjfm^l−1i−1,tl−1i−1−pjfm^l−1i−1,tl−1i−1+βjm^l−1i−1,tl−1i−1+∑l≠jnγj,lm^l−1i−1,tl−1i−1plfm^l−1i−1,tl−1i−1Δt+σj,jm^l−1i−1,tl−1i−1pjfm^l−1i−1,tl−1i−1Zj,jl+pjfm^l−1i−1,tl−1i−1∑l≠jnσj,lm^l−1i−1,tl−1i−1plfm^l−1i−1,tl−1i−1Zj,ll.It is clear that the 95% confidence interval for the forecast at timetk is
(96)yjfm^ki−1,tki−1−1.96sm∧k−1i−1,k−1j,jyjf1/2,yjfm^ki−1,tki−1+1.96sm∧k−1i−1,k−1j,jyjf1/2,pjfm^ki−1,tki−1−1.96sm∧k−1i−1,k−1j,jpjf1/2,pjfm^ki−1,tki−1+1.96sm∧k−1i−1,k−1j,jpjf1/2,where the lower ends denote the lower bounds of the state estimate and the upper ends denote the upper bounds of the state estimate.


## 8.1.2. Illustration: Prediction/Confidence Interval for Energy Commodities with No Jump

For the case of no jump, the following graphs show the simulated, forecasted, and 95 percent confidence limit for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively.Figures15(a)–15(c) show the graph of the forecast and 95 percent confidence limit for the case where there is no jump for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively. The figure shows two regions: the simulation region S and the forecast region F. For the simulation region S, we plot the real data set together with the simulated data set as described in Figure 14. For forecast region F, we plot the estimate of the forecast as explained in Section 8.1.1.Figure 15
Real, simulated, forecasted prices and 95% C.I. with no jump.
(a)
(b)
(c)


## 8.1.3. Illustration: Prediction/Confidence Interval for Energy Commodities with Jump

For the jump process, the following graphs show the forecast and 95 percent confidence limit for the daily Henry Hub natural gas data set [36], daily crude oil data set [37], and daily coal data set [35], respectively.Figures16(a)–16(c) show the graphs of the forecast and 95 percent confidence limit for the case where there is jump for the daily Henry Hub Natural gas data set [36], daily Crude Oil data set [37], and daily Coal data set [35], respectively. Figures 16(a)–16(c) show two regions: the simulation region S and the forecast region F. For the simulation region S, we plot the real data set together with the simulated data set as described in Figure 14. For the forecast region F, we plot the estimate of the forecast as explained in Section 8.1.1.Figure 16
Real, simulated, forecasted prices and 95% CI with jump.
(a)
(b)
(c)


## 9. Conclusion and Future Work

We examine the relationship between the prices of the energy commodities: natural gas, crude oil, and coal. To do this, multivariate stochastic models with and without external random interventions are developed to describe the relationship between the price of energy commodities. The time-varying parameters in the stochastic model are estimated using the LLGMM method. Figures7 and 14 suggest that the model with jump fits better than the model without jump. For j,l∈I1,3, the estimates for the drift interaction parameters γj,lm^k,tki−1 for the case where jump is incorporated suggest that there are definitely interactions between the spot price of these three commodities. As discussed in (57) and (58), the sign of these parameters suggest that there is competition or cooperation between commodities l and j. The estimate of the parameter γj,lm^k,tki−1 in Tables 3 and 10 and Figures 4 and 11 suggests that these commodities either compete or cooperate with each other depending on the time period. We can also describe the relationship between any two commodity j and l, j≠l∈I1,3, based on the overall average γ^j,l=1/N∑k=1Nγj,lm^k,tki−1. For example, for the case where jump is incorporated, γ^1,3=−0.0017 and γ^3,1=−0.0095. This suggests that on the average, there is competition between these two commodities. Also, γ^1,2=0.0018 and γ¯2,1=0.0083. This indicates that on the average, there is cooperation between natural gas and crude oil. Finally, γ^2,3=−0.0146 and γ^3,2=−0.0013. Therefore, on the average, there is competition between crude oil and coal. In the future, we plan to apply the local lagged adapted generalized method of moments to interconnected nonlinear stochastic dynamic model for log-spot price, expected log-spot price, and volatility process. Also, we plan to incorporate delay in the multivariate interconnected nonlinear stochastic model. We plan to be able to apply the extended local lagged adapted generalized method of moments to other multivariate interconnected nonlinear dynamic model different from energy commodity model.



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*Source: 2075258-2020-04-20.xml*