# Moment Equations in Modeling a Stable Foreign Currency Exchange Market in Conditions of Uncertainty

Authors: Josef Diblík; Irada Dzhalladova; Mária Michalková; Miroslava Růžičková
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The paper develops a mathematical model of foreign currency exchange market in the form of a stochastic linear differential equation with coefficients depending on a semi-Markov process. The boundaries of the domain of its instability is determined by using moment equations.


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## Body

## 1. Introduction

The economic growth of a given country is based on the government policy that includes numerous control moments. An important part of this policy is the correct financial policy, which defines the priorities in the development of financial relations and its function is to ensure the financial stability of the state. Finance and energy markets have been an active scientific field for some time, even though the development and applications of sophisticated quantitative methods in these areas are relatively new and referred to in a broader context as energy finance. Energy finance is often viewed as a branch of mathematical finance, yet this area continues to provide a rich source of issues that are fuelling new and exciting research developments [1].The foreign currency exchange market is one of the most liquid financial markets with banks as major participants. Income from the foreign currency exchange transactions makes up a significant proportion of the banks income. The currency exchange risks associated with open positions are especially imminent in periods of significant fluctuations in exchange rates. The main feature of risky cases related to the market risk is that such cases occur as a result of adverse changes in the general market situation. Whenever such cases occur, the value of the assets has a tendency to decrease for a short-term period, causing liquidity gap.In view of the disbalance of the foreign currency exchange market, the negative trade balance, the high inflation, an effective foreign currency exchange rate policy determining the optimal level of foreign currency exchange rates is an important problem.Under such conditions, it is especially important to perceive the “bank" as a comprehensive dynamic system that works in the conditions of unstable economy under high foreign currency exchange risks. Thus, a more widespread use of economic-mathematical methods and models is necessary to study the processes taking place in the “bank", evaluating the effectiveness of its work and identifying the trends and ways to improve the management of the banking activities.Significant scientific achievements in the field of banking and construction of some models can be found in [2, 3], and some economical models are studied in [4–6]. However, many other issues of bank practices require further research and elaboration of approaches to their solution. One aspect of the model is to build stable functioning of the foreign currency exchange transactions of the “bank" as a factor of effective functioning of the banking system in general [7, 8].Most scientists understand under the category “financial stability of the banking system” the establishment of an effective mechanism preventing the emergence of banking crises and facilitating further development of economy. Depending on the tasks, the stability of the banks may be defined as in the model presented in this paper.The paper develops a stability model of foreign currency bank transactions with semi-Markov fluctuations. An example illustrates the theory in the special case when the semi-Markov process can take three possible states. This means that a commercial bank operates in a foreign currency exchange market that can be in three states: stable foreign currency exchange market, market in the crisis, and market with currency restrictions. In the example, we assume that the bank remains in each state for the same period of time.In addition, the present paper contains the necessary and sufficient conditions for the mean square stability and conditions for theL2-stability of systems with semi-Markov coefficients and random transformations of solutions. There are constructed moment equations as a tool for studying the stochastic system stability which is working in uncertainty conditions.


## 2. Statement of the Problem

Let(Ω,ℱ,F,ℙ) be a filtered probability space (or stochastic basis) consisting of a probability space (Ω,ℱ,ℙ) and a filtration
(1)F={ℱt,∀t≥0}⊂ℱ.The spaceΩ is called the sample space, ℱ is the set of all possible events (the σ-algebra), and ℙ is some probability measure on Ω. A family ξ={ξ(t):t≥0} of random variables ξ(t):Ω→𝕃2 is called a continuous-time stochastic process on the state space 𝕃2. In our considerations, ξ(t) is a random semi-Markov process and the state space 𝕃2 is the space of all random variables for which there exists squared mathematical expectation. On such a probability space, we consider initial problem formulated for the stochastic system of linear differential equations in the form
(2)dx(t)dt=A(ξ(t))x(t),(3)x(0)=φ(ω),
where A(ξ(t)) is an m×m matrix whose elements depend on the semi-Markov process ξ(t). The state function x(t) is an m-dimensional column vector-function with the initial state φ(ω) at t=0. For simplicity, we denote
(4)A(ξ(t))=Ak,ifξ(t)=θk,k=1,2,…,n,
where Ak are constant m×m matrices and θk are real numbers.Anm-dimensional vector-function x(t) is called a solution of the initial value problem (2) and (3) if x(t) satisfies (2) and initial condition (3) within the meaning of a strong solution of the initial Cauchy problem.Our considerations are subject to the following assumptions.Assumption 1.
The random semi-Markov processξ(t) can take n possible states
(5)θ1,θ2,…,θn
with transition probabilities
(6)πsk(t)=P{ξ(tj+1)=θs∣ξ(tj)=θk,j=1,2,…},=θs∣ξ(tj)=θk,j=1,2,…s,k=1,2,…,n,
where t0=0<t1<t2<⋯ are the moments of time at which the jump from one state to another is realized.
If we fix any momentt>0, then the semi-Markov process takes some of states, ξ(t)=θk, k=1,2,…,n, and the state function x(t)≡x(t,ω) changes in accordance with the deterministic system of differential equations
(7)dx(t)dt=Akx(t),k=1,2,…,n.
So the solution of such a system is in the form
(8)x(t)=eAktφ(ω),k=1,2,…,n.Assumption 2.
The jumping time during which the process is in stateθs before it jumps to state θk, s,k=1,2,…,n is given by a discrete integer-valued random variable Tsk whose probability density function is a known function dsk(t). Then, the intensity qsk(t) of the jump from state θs to state θk is given by the formula
(9)qsk(t)=πsk(t)dsk(t),s,k=1,2,…,n,
and the semi-Markov process ξ(t) is defined by the transition intensity matrix
(10)Q(t)=(qsk(t))s,k=1n,
whose elements satisfy the relationships
(11)qsk(t)≥0,∫0∞‍∑s=1n‍qsk(t)dt=∫0∞qk(t)dt=1,
where qk(t) is the probability density of the elapsed time Tk in state θk if the process jumps to it at time tj. If ψk(t) denotes the probability of the event that no jump takes place during the interval (tj,tj+1), provided that the process jumps to the state θk at time tj, then
(12)ψk(t)=∫t∞qk(τ)dτ,k=1,2,…,n.
In our considerations, it will be convenient to denote the block-diagonal matrix,
(13)Ψ(t)=diag(ψ1(t),ψ2(t),…,ψn(t)).Assumption 3.
At the moments of jumpstj,j=1,2,… that are caused by some perturbations, solutions of (2) submit to the random transformations
(14)x(tj+0)=Cskx(tj-0),s,k=1,…,n,
where Csk are m×m constant matrices and detCsk≠0.Our aim is to transform the stochastic system with random coefficients to a deterministic system with solutions whose stability can be considered by using classical methods. To complete this task below, we present a method of moment equations. We will show that the method is effective and useful for solving an economical model problem.


## 3. Construction of the Moment Equations

We define the moments of the first or second order of a random variablex before we derive the moments equations. We use some notation. In the sequel, 𝔼m denotes an m-dimensional Euclidean space, functions fk(t,x), k=1,2,…,n are the particular density functions of the random variable x, and the vector-function
(15)f(t,x)=(f1(t,x),f2(t,x),…,fn(t,x))T,
where the operation T denotes transposition, is called the vector of particular density functions.  Moreover, we define
(16)S(t):=(qskSsk)s,k=1n,s,k=1,…,n,R(t):=diag(R1(t),…,Rn(t)),
where Ssk are operators defined, for a given function f, as
(17)Sskf(t,x)≡fk(t,Csk-1x)detCsk-1
and Rk, k=1,…,n are operators defined, for a given function f, as
(18)Rk(t)f(t,x)≡fk(t,e-Aktx)det(e-Akt).Definition 4.
Letx∈𝔼m be a continuous random variable depending on a random semi-Markov process ξ(t) with n possible states θk, k=1,2,…,n. The n-dimensional column vectors E(1){x(t)} and n×n matrices E(2){x(t)} of the form
(19)E(1){x(t)}=∑k=1n‍Ek(1){x(t)},E(2){x(t)}=∑k=1n‍Ek(2){x(t)},
where
(20)Ek(1){x(t)}=∫𝔼mxfk(t,x)dx,Ek(2){x(t)}=∫𝔼mxxTfk(t,x)dx,
are called moments of the first or second order of the random variable x, respectively. The values Ek(1){x(t)} and Ek(2){x(t)}, k=1,…,n are called particular moments of the first or second order, respectively.Theorem 5.
Let the coefficients of the linear differential system (2) depend on a random semi-Markov process ξ(t) with transition intensity matrix (10) and, for solutions of system (2), there occur jumps (14) simultaneously with jumps of the process ξ(t). Then, the following three statements are true.
(
1
) The stochastic process (x(t),ξ(t)) is defined by the operator equation
(21)f(t,x)=L(t)f(0,x),
where the matrix operator L(t)≡(Lij(t))i,j=1n satisfies
(22)L(t)=ψ(t)R(t)+∫0t‍L(t-τ)S(τ)R(τ)dτ.
(
2
) The vectors of particular moments of first order satisfy
(23)Ek(1){x(t)}=ψk(t)eAktEk(1){x(0)}+∫0tψ(t-τ)eAk(t-τ)Zk(τ)dτ,
where
(24)Zk(t)=∑s=1n‍qks(t)CkseAstEs(1){x(0)}+∫0t‍∑s=1n‍qks(t-τ)CkseAs(t-τ)Zs(τ)dτ
and k=1,…,n.
(
3
) The matrix of particular moments of second order satisfies
(25)Ek(2){x(t)}=ψk(t)eAktEk(2){x(0)}eAkTt+∫0tψk(t-τ)eAk(t-τ)Wk(τ)eAkT(t-τ)dτ,
where
(26)Wk(t)=∑s=1n‍qks(t)CkseAstEs(2){x(0)}eAsTtCksT+∫0t‍∑s=1n‍qks(t-τ)CkseAs(t-τ)Ws(τ)eAsT(t-τ)CksTdτ,
and k=1,…,n.Proof.
(
1
)  The stochastic process (x(t),ξ(t)) is also semi-Markov because all probabilistic properties of the process for t>tj are defined by particular probability density functions at the moment tj of jump. Thus, there exists a linear operator L(τ) such that
(27)f(tj+τ,x)=L(τ)f(tj,x),j=0,1,2,….
Let the stochastic process ξ(t) move to state θk at the moment t=0. Then,
(28)pk(0)=1,ps(0)=0,fors≠k,s=1,…,n,(29)fk(0,x)≥0,∫Em‍fk(0,x)dx=1,fs(0,x)≡0,s≠k.
For particular density functions, when t≥0, we obtain
(30)fs(t,x)=Lsk(t)fk(0,x),k,s=1,…,n.
Also, with probability ψk(t), in view of x(0)=e-Aktx, we obtain the equality
(31)fk(t,x)=fk(0,e-Aktx)det(e-Akt),fs(t,x)≡0,2222222222222222222222222222222222k≠s.
On the interval (τ,τ+dτ), there could be a jump of the stochastic process ξ(τ) from state θk to state θs with probability qks(τ)dτ. Taking into account that functions qks(t) are continuous, we obtain the probability
(32)P{τ<t1<τ+dτ}=∫ττ+dτ‍qk(s)ds≈qk(τ)dτ.
After a jump at the moment lying between moments τ and τ+dτ, we can use (27). At the moment t1 of the first jump of the stochastic process ξ(t), in accordance with (14), we have
(33)fs(t1+0,x)=fk(t1-0,Csk-1x)detCsk-1.
Therefore, by using operators R,S defined in (16), to remain in state θk at the moment t, we get
(34)Lkk(t)fk(0,x)=ψk(t)Rk(t)fk(0,x)+∫0t‍∑s=1n‍Lks(t-τ)qsk(τ)SskRk(τ)fk(0,x)dτ,Lks(t-τ)qsk(τ)SskRk(τ)fk(0,x)d2τ,2k=1,…,n.
For transition from state θk to state θτ at the moment t, we obtain
(35)Lτk(t)fk(0,x)=∫0t‍∑s=1n‍Lτs(t-τ)qsk(τ)SskRk(τ)fk(0,x)dτ,
where τ≠k, τ,k=1,…,n.
These two systems can be written in the form (22).
(
2
)  Before we derive the system of moment equations in (23), we establish an auxiliary operator equation. Let us find the solution of (22) in the form
(36)L(t)=ψ(t)R(t)+∫0t‍ψ(t-τ)R(t-τ)U(τ)dτ,
where U is an unknown matrix. If we put L(t) expressed in the form (36) into (22), we obtain
(37)∫0t‍ψ(t-τ)R(t-τ)U(τ)dτ=∫0tψ(t-τ)T(t-τ)S(τ)R(τ)dτ+∫0tdτ∫0t-τ‍ψ(t-τ-s)R(t-τ-s)U(s)S(τ)R(τ)ds.
After substituting r=τ+s, τ=τ in the double integral and after changing the order of integration, we obtain
(38)∫0t‍dτ∫0t-τ‍ψ(t-τ-s)R(t-τ-s)U(s)S(τ)R(τ)ds=∫0t‍ψ(t-r)(∫0r‍U(r-τ)S(τ)R(τ)dτ)dr.
Therefore, a suitable matrix U in (22) is the matrix
(39)U(t)=S(t)R(t)+∫0t‍U(t-τ)S(τ)R(τ)dτ.
In the other way, we can find a matrix U(t) as a solution of (36) in the form
(40)U(t)=S(t)R(t)+∫0t‍S(t-τ)R(t-τ)V(τ)dτ,
where V is an unknown matrix. From this, we get
(41)V(t)=S(t)R(t)+∫0t‍V(t-τ)S(τ)R(τ)dτ.
A comparison of (39) and (41) implies that we can set V(t)≡U(t). Then, (40) can be written as
(42)U(t)=S(t)R(t)+∫0t‍S(t-τ)R(t-τ)U(τ)dτ,
or
(43)U(t)=S(t)R(t)+∫0t‍S(τ)R(τ)U(t-τ)dτ.
Multiplying (36) and (38) on the right by the vector f(0,x), we obtain
(44)f(t,x)=ψ(t)R(t)f(0,x)+∫0t‍ψ(t-τ)R(t-τ)U(τ)f(0,x)dτ,U(t)f(0,x)=S(t)R(t)f(0,x)+∫0t‍S(t-τ)R(t-τ)U(τ)f(0,x)dτ.Denote (45)h(t,x)=U(t)f(0,x),h(t,x)=(h1(t,x),h2(t,x),…,hn(t,x))T,and (46)zk(t)=∫𝔼mxhk(t,x)dx,Wk(t)=∫𝔼mxxThk(t,x)dx,
where k=1,…,n. The system (44) can be rewritten into the scalar form
(47)fk(t,x)=ψk(t)Rk(t)fk(0,x)+∫0t‍ψk(t-τ)Rk(t-τ)hk(τ,x)dτ,(48)hk(t,x)=∑s=1n‍qks(t)SksRs(t)fs(0,x)+∫0t‍∑s=1n‍qks(t-τ)SksRs(t-τ)hs(τ,x)dτ,qks(t-τ)SksRs(t1-τ)hs(τ,x),k=1,…,n.
The system of (23) can be obtained by multiplying each equation of (48) by vector x and integrating it over the space 𝔼m. In doing so, it is necessary to use
(49)∫𝔼mxfk(t,x)dx=Ek(1){x(t)},∫𝔼mxRk(t)fk(0,x)=∫𝔼mxfk(0,e-Aktx)dete-Aktdx=∫𝔼meAktyfk(0,y)dy=eAktEk(1){x(t)},∫𝔼mxRk(t-τ)hk(τ,x)dx=∫𝔼mxhk(τ,e-Ak(t-τ))dete-Ak(t-τ)dx=eAk(t-τ)∫𝔼myhk(τ,y)dy=eAk(t-τ)zk(τ),∫𝔼mxSksRk(t)fs(0,x)dx=∫𝔼mxSksfs(0,e-Astx)dete-Astdx=∫𝔼mxfs(0,e-AstCks-1x)dete-AstCks-1dx=CkseAst,∫𝔼mxfs(0,x)dx=CkseAstEs(1){x(0)}.
(
3
) The system of (25) can be obtained by multiplying each equation in (48) by matrix xxT and integrating it over the space 𝔼m by using matrix equalities
(50)∫𝔼m‍xxTRk(t)fk(0,x)dx=∫𝔼m‍xxTfk(0,e-Aktx)dete-Aktdx=∫𝔼m‍eAktyyTeAkTtfk(0,y)dy=eAktEk(2){x(0)}eAkTt,∫𝔼m‍xxTSksRs(t)fs(0,x)dx=∫𝔼m‍xxTfs(0,e-AktCks-1x)dete-AktdetCks-1dx=CkseAstEs(2){x(0)}eAsTtCksT.


## 4. Necessary and Sufficient Conditions ofL2-Stability

Several different stability definitions are useful. Here, we recall the mean stability and the mean square stability definitions, theL2-stability, and the classical definition of asymptotic stability.Definition 6.
The trivial solution of system (2) is said to be mean square stable on the interval [0,∞) if, for each ε>0, there exists δ>0 such that any solution x(t) corresponding to the initial data x(0) exists for all t≥0 and the mathematical expectation
(51)E(1){∥x(t)∥2}<ε,whenevert≥0,∥x(0)∥<δ.The mean stability of the zero solution of system (2) is much defined in the same way with only ∥x(t)∥2 being replaced by ∥x(t)∥.Definition 7.
The trivial solution of system (2) is said to be asymptotically mean square stable on the interval [0,∞) if it is stable and, moreover,
(52)limt→∞E(2){x(t)}=0.Remark 8.
It is obvious that the mean stability of the zero solution of system (2) is equivalent to the asymptotic stability of the solutions of system (23) and (24) and the mean square stability of the solutions of system (2) is equivalent to the asymptotic stability of the solutions of system (25) and (26).Definition 9.
The trivial solution of the differential systems (2) is said to be L2-stable if the integral
(53)∫0∞E(1){∥x(t)∥2}dt
converges.Remark 10.
It is easy to see that the integral (53) converges if and only if the matrix integral
(54)∫0∞E(2){x(t)}dt
is convergent.Lemma 11.
The following three inequalities hold:(1)
Ek(2){x(t)}≥0,k=1,…,n.(2)
eAktEk(2){x(0)}eAkTtx≥0,k=1,…,n.(3)
CkseAstEs(2){x(0)}eAsTtCksT≥0,k,s=1,…,n.Proof.
All inequalities follow from propertyfk(t,x)≥0, k=1,…,n in accordance with (23) and (24).It is convenient to derive some sufficient and necessary conditions ofL2-stability by using a matrix operator with suitable properties. Such an operator is defined by the following lemma.Lemma 12.
The linear matrix operatorsNks(t), k,s=1,…,n, defined as
(55)Nks(t)∘Ws(t):=∫0t‍qks(t-τ)CkseAs(t-τ)Ws(τ)eAsT(t-τ)CksTdτ,
are monotonous.Proof.
Becauseqks(t-τ)≥0, in accordance with the third statement of Lemma 11, we have
(56)xTNks(t)∘Ws(t)x≥0,forx≥0.
So Ws(τ)≥0 implies Nks(t)∘Ws(t)≥0 and the operator is monotone.Remark 13.
The linear monotonous operatorNks(t) transforms any symmetric matrix D into a symmetric matrix Nks(t)∘D. Its monotonicity guarantees that inequality D1≥D2 implies inequality
(57)Nks(t)∘D1≥Nks(t)∘D2.Lemma 14.
The symmetric matricesWk(t),k=1,…,n defined by (26) satisfy the inequalities Wk(t)≥0, k=1,…,n.Proof.
System (26) can be rewritten in the form
(58)Wk(t)=∑s=1n‍qks(t)CkseAstEs(2){x(0)}eAsTtCksT+∑s=1n‍Nks(t)∘Ws(t),k=1,…,n, by using the matrix operators (55). System (26) as well as system (58) can be solved by the method of successive approximations
(59)Wk(0)(t)=∑s=1n‍qks(t)CkseAstEs(2){x(0)}eAsTtCksT,Wk(l+1)(t)=Wk(0)(t)+∑s=1n‍Nks(t)∘Ws(l)(t),l=1,2,….
Hence, Wk(0)(t)≥0 and Wk(l)(t)≥0 implies Wk(l+1)≥0, k=1,…,n. So the solution of system (58),
(60)Wk(t)=liml→∞Wk(l)(t),
satisfies Wk(t)≥0, k=1,…,n.Now, we rewrite the moment equations (25) and (26) into a more compact form by using the denotations
(61)Dk=∫0∞Ek(2){x(t)}dt,Wk=∫0∞Wk(τ)dτ,k=1,…,n.
Then, integrating systems (25) and (26) from 0 to ∞ with respect to t, we obtain
(62)Dk=∫0∞ψk(t)eAkt(Ek(2){x(0)}+Wk)eAkTtdt,k=1,…,n,(63)Wk=∑s=1n‍∫0∞qks(t)CkseAst(Es(2){x(0)}+Ws)eAsTtCksTdt,k=1,…,n.Corollary 15.
Let the zero solution of the system (2) with jumps (14) at random time moments tj,j=0,1,2,… determined by jumps of stochastic process ξ(t) be L2-stable. Then, the integrals
(64)Ik=∫0∞ψk(t)eAktEk(2){x(0)}eAkTtdt,k=1,…,n
are convergent.Proof.
This immediately follows from Lemma14. In fact, since Wk≥0, then Dk≥Ik, k=1,…,n.Theorem 16.
Let the integrals (64) be convergent. Then, the zero solution of system (2) is L2-stable if and only if the solutions Wk≥0, k=1,…,n of system (63) are bounded.Proof.
(
1
)
Sufficiency. Integrating system (59) from 0 to ∞ with respect to t, using notation Wk(l)=∫0∞‍Wk(l)(t)dt, k=1,…,n, we obtain the system of matrix successive approximations
(65)Wk(l+1)=Wk(0)+∑s=1n‍Nks∘Ws(l),k=1,…,n,l=0,1,2,….
As the linear operators Nks, k,s=1,…,n defined by (55) are monotonous, the zero solution of system (2) is L2-stable if the sequence of matrices Wk(l), l=0,1,2,… is convergent.
(
2
)
Necessity. Let us assume that the solution Wk=Zk≥0, k=1,…,n of the system
(66)Wk=Wk(0)+∑s=1n‍Nks∘Ws,k=1,…,n
is bounded. Obviously, Zk≥Wk(0), k=1,…,n and, in view of successive approximations (65), we get
(67)Wk(l+1)=Wk(0)+Nks∘Ws(l)≤Wk(0)+Nks∘Zs=Zk,k=1,…,n.
So, for all l=0,1,2,…, we have Wk(l)≤Zk, k=1,…,n.
Next, from (65), we obtain
(68)Wk(l+1)=Wk(0)+Nks∘Ws(l)≥Wk(0)+Nks∘Ws(l-1)=Wk(l),k=1,…,n.
Moreover, becauseWk(0)≥0, Wk(l+1)≥Wk(l), k=1,…,n is satisfied for all l=0,1,….
Finally, the boundedness and monotonicity of the matrix sequencesWk(l), l=0,1,… imply the existence of limits
(69)Wk=liml→∞Wk(l),k=1,…,n. Consequently, independently of the initial value Wk(0), in view of 0≤Wk(l)≤Zk, the linear operator N=(Nks)k,s=1n has the spectral radius less than 1. This means that system (65) has a unique solution Wk=Zk, k=1,…,n.


## 5. Model Problem

The functioning of the foreign currency exchange market in conditions of uncertainty can be modelled by using stochastic differential equations. Such convenient mathematical model is the scalar case of the initial problem (2), (3), that is, the initial problem formulated for the stochastic linear differential equation
(70)dx(t)dt=a(ξ(t))x(t),(71)x(0)=φ(ω),
where coefficient a depends on a semi-Markov process ξ(t). The possible states θ1,…,θn of the stochastic process ξ(t) express the conditions in which the bank works, for example, in a currency crisis, in a stable foreign currency exchange market, and so forth. Let the stochastic process ξ(t) take the states θk, k=1,2,…,n. If ξ(t)=θk, we denote a(ξ(t))=ak. Further assume that the intensities qsk(t) are determined by the formulas
(72)qss(t)=0,(73)qsk(t)={1Tskfor0≤t<Tsk,0fort≥Tsk,
where s,k=1,2,…,n.Perturbations in the foreign currency exchange market cause the changes of the stochastic processξ(t), and consequently, solutions of (2) in this scalar case are subject to the random transformations(74)x(tj+0)=pkx(tj-0),pk≠0,k=1,…,n
at the moments of jumps tj, j=1,2,….We derive the domain of stability of the foreign currency exchange market using the results of the previous section. The moment equations in (25) for Ek(2){x(t)}, k=1,…,n take, in the scalar case, the form
(75)Ek(2){x(t)}=ψke2aktEk(2){x(0)}+∫0t‍ψk(t-τ)e2ak(t-τ)Wk(τ)dτ,(76)Wk(t)=∑s=1n‍qks(t)Cks2e2astEs(2){x(0)}+∫0t‍∑s=1n‍qks(t-τ)Cks2e2as(t-τ)Ws(τ)dτ,qks(t-τ)Cks2e2as(t-τ)Ws(τ)dτ,k=1,…,n.
By definition, the zero solution of (75) is asymptotically stable if E(2){x(t)}→0 for t→∞.Lemma 17.
If(77)limt→∞Ek(2){x(t)}=0,k=1,…,n,
then
(78)limt→∞ψk(t)e2akt=0,k=1,…,n.Proof.
Formula (20) implies Ek(2){x(t)}≥0, k=1,…,n if t≥0. Therefore, Wk(t)≥0,k=1,…,n,t≥0. From (75), it follows that inequalities
(79)Ek(2){x(t)}≥ψke2aktEk(2){x(0)},k=1,…,n
are always satisfied. Then, for any constant Ek(2){x(0)}, property (77) implies (78).Note that, in general, condition (78) does not imply the property (77). In the following theorem, it is shown what additional assumptions are needed.Theorem 18.
Let (78) hold and let
(80)∫0∞‍ψk(t)e2aktdt<∞,k=1,…,n.
Then, if there exist limits
(81)limt→∞Wk(t)=0,k=1,…,n,
limits (77) exist too.Proof.
We denote(82)Ik=∫0∞‍ψk(t)e2astdt,Ik≥0,k=1,…,n.
As the integrals Ik, k=1,…,n are convergent, for all ε>0, there exists T1>0 such that, for all t>T1, the inequalities
(83)∫t∞‍ψk(t)e2akt<ε,k=1,…,n
hold. Similarly, assumption (81) means that ∀ε>0∃T2>0 such that ∀t>T2 the inequalities
(84)0≤Wk(t)<ε,k=1,…,n
hold. Moreover, there exists constant W0 such that
(85)|Wk(t)|≤W0,k=1,…,n,t≥T2.
The integral part of (75) can be now estimated if t>T1+T2. We have
(86)∫0t‍ψk(t-τ)e2ak(t-τ)Wk(τ)dτ=∫0T1+T2ψk(t)e2akτWk(t-τ)dτ+∫T1+T2tψk(t)e2akτWk(t-τ)dτ≤Ikε+W0ε.
Thus, there exist limits
(87)limt→∞∫0t‍ψk(t-τ)e2ak(t-τ)Wk(τ)dτ=0,k=1,…,n.Corollary 19.
Let the assumptions of Theorem18 hold. Then, the asymptotical stability of solutions of system (75) implies the asymptotical mean square stability of the zero solution of (70).Proof.
Under the given assumptions, this follows from the existence of limits (77).The results obtained make it possible to examine the stability of the stochastic equation (70) by using the deterministic system of (76). Here, we can use the known methods such as the Laplace transformation.If we denote(88)Υk(p)=∫0∞‍e-ptWk(t)dt,Θks(p)=∫0∞‍e-ptqks(t)dt,(t)dt,Θks(p)=∫0∞‍e-pt222qks(t)k,s=1,…,n,
then, multiplying (76) by e-pt and integrating it from 0 to ∞ with respect to t, (76) can be transformed into the system of linear algebraic equations with respect to the functions Υk(p). We get the system
(89)Υk(p)=∑s=1n‍Θks(p-2as)Cks2Es(2){x(0)}+∑s=1n‍Θks(p-2as)Cks2Υs(p),
where k=1,…,n.The determinantΔ(p) of the system of linear equations (89) is in the form(90)Δ(p)=|1-Θ11(p-2a1)C112-Θ12(p-2a2)C122⋯-Θ1n(p-2an)C1n2-Θ21(p-2a1)C2121-Θ22(p-2a2)C222⋯-Θ2n(p-2an)C2n2⋮⋮⋱⋮-Θn1(p-2a1)Cn12-Θn2(p-2a2)Cn22⋯1-Θnn(p-2an)Cnn2|. The singular points of mappings Θks(p), k,s=1,…,n are determined by the roots of Δ(p)=0. If all the functions Θks(p), k,s=1,…,n are entire, then there are no singular points, except for the point p=∞.Remark 20.
In a particular case, solutions of (89) are located on the boundary of the stability domain. If p=0, then Δ(0)=0 is the equation determining the boundary of the stability domain.To solve the model problem formulated at the beginning of this section, we use the stochastic operatorsSskf(x), s,k=1,2,…,n in the form
(91)Sskf(t,x)={f(t,x),s=k,∑k=1n‍pkpsf(t,xps),s≠k,
associated to the intensities qsk(t) determined by (72). The domain of stability of banking operations in the foreign currency exchange market can be derived from the behavior of the solutions of the moment equations (76). Before we use the moment equations, we express the probabilities Cks, k,s=1,2,…,n from formula (14) by using the probabilities pk, k=1,2,…,n from formula (74) in the form
(92)Cks={1,s=k,∑k=1n‍pk2,s≠k.
Then, the moment equations (76) can be rewritten into the form
(93)Wk(t)=ρ[∑s=1s≠knqks(t)e2astEs(2){x(0)}+∫0t‍∑s=1n‍qks(t-τ)e2as(t-τ)Ws(τ)dτ∑s=1s≠knqks(t)e2astEs(2){x(0)}],qks(t-τ)e2as(t-τ)Ws(τ)2k=1,…,n,
where ρ=∑k=1npk2. The system (93) can be solved in the same way as above, that is, by Laplace transformation. Then, multiplying (93) by e-pt and integrating it from 0 to ∞ with respect to t, we get a preliminary form of the system
(94)Υk(p)=ρ[∫0∞‍∑s=1k≠snqkse2aste-ptEs(2){x(0)}dt+∫0∞‍∫0t‍∑s=1k≠snqks(t-τ)e2as(t-τ)e-ptWs(τ)dτdt],∑s=1k≠sn‍qks(t-τ)444e2as(t-τ)e-pt22Ws(τ)d22τk=1,…,n,
which we will still have to modify using some properties of the Laplace transformation. In accordance with the property of the delay and with regard to the equality
(95)∫0∞qkse-ptdt=1-e-pTksTksp,k,s=1,…,n,k≠s,
we obtain
(96)∫0∞qkse2aste-ptdt=1-e-Tks(p-2as)Tks(p-2as),k,s=1,…,n,k≠s.
In accordance with the property of convolution, we get
(97)∫0∞∫0tqks(t-τ)e2as(t-τ)e-ptWs(τ)dτdt=1-e-Tks(p-2as)Tks(p-2as).
Therefore, the system of (94) can be written in the form
(98)Υk(p)=ρ[∑s=1k≠snEs(2){x(0)}1-e-Tks(p-2as)Tks(p-2as)+∑s=1k≠snfs(p)1-e-Tks(p-2as)Tks(p-2as)],
where k=1,…,n, or, using the notations
(99)bk(p)≡∑s=1k≠snEs(2){x(0)}1-e-Tks(p-2as)Tks(p-2as),aks≡1-e-Tks(p-2as)Tks(p-2as),k,s=1,…,n, k≠s, in a simpler form
(100)Υk(p)=ρ(bk(p)+∑s=1k≠snaksΥs(p)),k=1,…,n.By the Cramer theorem, we can solve the system (100). The singular points are determined by the roots of
(101)det(E-ρa)=0,
where a≡(aks(p,Tks))k,s=1n, while ρakk(p,Tkk)≡-1.The character of the roots of (101) determines the stability of the solutions of the system of integral equations in (93). If the real parts of all the roots of (101) are negative, then the solutions of (93) are asymptotically stable. If there is at least one root of (101) with a positive real part, then the solutions of integral equations (93) are unstable.The character of the dependence between parametersp and Tks can be determined by solving the system of algebraic equations in (101) by numerical methods.Example 21.
The real boundaries of the instability domain of foreign currency exchange market can be determined in a particular case. Suppose that the random semi-Markov process can take three states:θ1—if the bank operates in a currency crisis, then a(ξ(t))=a1;θ2—if the bank operates in a stable foreign currency exchange market, then a(ξ(t))=a2;θ3—if the bank operates in a market with currency restrictions, then a(ξ(t))=a3,with intensities
(102)q11(t)=q22(t)=q33(t)≡0,q12(t)=q13(t)=q21(t)=q23(t)=q31(t)=q32(t)≡{1Tfor0≤t<T,0fort>T,
which means that the bank remains in each state for the same period of time 1/T. In the above case, the system (100) takes the form
(103)Υ1(p)=ρb1(p)+ρa12Υ2(p)+ρa13Υ3(p),Υ2(p)=ρb2(p)+ρa21Υ1(p)+ρa23Υ3(p),Υ3(p)=ρb3(p)+ρa31Υ1(p)+ρa32Υ2(p),
where
(104)a12=1-e-T(p-2a2)T(p-2a2),a13=1-e-T(p-2a3)T(p-2a3),a21=1-e-T(p-2a1)T(p-2a1),a23=1-e-T(p-2a3)T(p-2a3),a31=1-e-T(p-2a1)T(p-2a1),a32=1-e-T(p-2a2)T(p-2a2).
The value ρ expresses the mean value of the bank's income from foreign currency transactions during time T. The singular point is p=0, when the solution is situated on the boundary of the domain of instability on the plane of coefficients a1, a2, and a3.
Equation (101) is in the form
(105)|1-ρa12-ρa13-ρa211-ρa23-ρa31-ρa321|=0,
or
(106)1-ρ3a21a32a13-ρ3a12a23a31-ρ2a13a31-ρ2a23a32-ρ2a12a21=0.Ifp=0, the boundaries of instability of solutions of (70) are constructed in the plane of parameters a1,a2, and a3 for different values ρ (see Figure 1 where some admissible boundaries are constructed).The boundary of instability of solutions of (70) constructed in the plane of parameters a1, a2, and a3 for p=0 and for different values of ρ.
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