# Diffusion Approximations of the Geometric Markov Renewal Processes and Option Price Formulas

Authors: Anatoliy Swishchuk; M. Shafiqul Islam
Journal: International Journal of Stochastic Analysis
 (2010)

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## Abstract

We consider the geometric Markov renewal processes as a model for a security
market and study this processes in a diffusion approximation scheme. Weak convergence
analysis and rates of convergence of ergodic geometric Markov renewal processes in diffusion
scheme are presented. We present European call option pricing formulas in the case of
ergodic, double-averaged, and merged diffusion geometric Markov renewal processes.


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## Body

## 1. Introduction

LetN(t) be a standard Poisson process and (Yk)k∈Z+ be i.i.d. random variables which are independent of N(t) and S0*>0. The geometric compound Poisson processes(1.1)St*=S0*∏k=1N(t)(1+Yk),t>0,
is a trading model in many financial applications with pure jumps [1, page 214]. Motivated by the geometric compound Poisson processes (1.1), Swishchuk and Islam [2] studied the Geometric Markov renewal processes (2.5) (see Section 2) for a security market in a series scheme. The geometric Markov renewal processes (2.5) are also known as a switched-switching process. Averaging and diffusion approximation methods are important approximation methods for a switched-switching system. Averaging schemes of the geometric Markov renewal processes (2.5) were studied in [2].The singular perturbation technique of a reducible invertible-operator is one of the techniques for the construction of averaging and diffusion schemes for a switched-switching process. Strong ergodicity assumption for the switching process means that the singular perturbation problem has a solution with some additional nonrestrictive conditions. Averaging and diffusion approximation schemes for switched-switching processes in the form of random evolutions were studied in [3, page 157] and [1, page 41]. In this paper, we introduce diffusion approximation of the geometric Markov renewal processes. We study a discrete Markov-modulated (B,S)-security market described by  a geometric Markov renewal process (GMRP). Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes in diffusion scheme are presented. We present European call option pricing formulas in the case of ergodic, double-averaged, and merged diffusion geometric Markov renewal processes.The paper is organized as follows. In Section2 we review the definition of the geometric Markov renewal processes (GMRP) from [2]. Moreover we present notation and summarize results such as random evolution of GMRP, Markov renewal equation for GMRP, infinitesimal operator of GMRP, and martingale property of GMRP. In Section 3 we present diffusion approximation of GMRP in ergodic, merged, and double-averaging schemes. In Section 4 we present proofs of the above-mentioned results. Section 4 contains solution of martingale problem, weak convergence, rates of convergence for GMRP, and characterization of the limit measure. In Section 5 we present merged diffusion GMRP in the case of two ergodic classes. European call option pricing formula for ergodic, merged, and diffusion GMRP are presented in Section 6.


## 2. The Geometric Markov Renewal Processes (GMRP)

In this section we present the Geometric Markov renewal processes. We closely follow [2].Let(Ω,ℬ,ℱt,ℙ) be a standard probability space with complete filtration ℱt and let (xk)k∈ℤ+ be a Markov chain in the phase space (X,𝒳) with transition probability P(x,A), where x∈X,A∈𝒳. Let (θk)k∈ℤ+ be a renewal process which is a sequence of independent and identically distributed (i.i.d.) random variables with a common distribution function F(x):=ℙ{w:θk(w)≤x}. The random variables (θk)k∈ℤ+ can be interpreted as lifetimes (operating periods, holding times, renewal periods) of a certain system in a random environment. From the renewal process (θk)k∈ℤ+ we can construct another renewal process (τk)k∈ℤ+ defined by(2.1)τk:=∑n=0kθn.
The random variables τk are called renewal times (or jump times). The process(2.2)v(t):=sup⁡{k:τk≤t}
is called the counting process.Definition 2.1 (see [1, 4]).
A homogeneous two-dimensional Markov chain(xn,θn)n∈ℤ+ on the phase space X×ℝ+ is called a Markov renewal process (MRP) if its transition probabilities are given by the semi-Markov kernel
(2.3)Q(x,A,t)=P{xn+1∈A,θn+1≤t∣xn=x},∀x∈X,A∈X,t∈R+.Definition 2.2.
The process(2.4)x(t):=xv(t)
is called a semi-Markov process.The ergodic theorem for a Markov renewal process and a semi-Markov process respectively can be found in [3, page 195], [1, page 66], and [4, page 113].Let(xn,θn)n∈ℤ+ be a Markov renewal process on the phase space X×ℝ+ with the semi-Markov kernel Q(x,A,t) defined in (2.3), and let x(t):=xv(t) be a semi-Markov process where the counting process v(t) is defined in (2.2). Let ρ(x) be a bounded continuous function on X such that ρ(x)>-1. We define the geometric Markov renewal process (GMRP) {St}t∈ℝ+ as a stochastic functional St defined by(2.5)St:=S0∏k=1v(t)(1+ρ(xk)),t∈R+,
where S0>0 is the initial value of St. We call this process (St)t∈ℝ+ a geometric Markov renewal process by analogy with the geometric compound Poisson processes(2.6)St*=S0*∏k=1N(t)(1+Yk),
where S0*>0,  N(t) is a standard Poisson process, (Yk)k∈Z+ are i.i.d. random variables. The geometric compound Poisson processes {St*}t∈ℝ+ in (2.6) is a trading model in many financial applications as a pure jump model [5, 6]. The geometric Markov renewal processes {St}t∈ℝ+ in (2.5) will be our main trading model in further analysis.Jump semi-Markov random evolutions, infinitesimal operators, and Martingale property of the GMRP were presented in [2]. For the convenience of readers we repeat them again in the following.


### 2.1. Jump Semi-Markov Random Evolutions

LetC0(ℝ+) be the space of continuous functions on ℝ+ vanishing at infinity, and let us define a family of bounded contracting operators D(x) on C0(ℝ+) as follows:(2.7)D(x)f(s):=f(s(1+ρ(x)),x∈X,s∈R+.
With these contraction operators D(x) we define the following jump semi-Markov random evolution (JSMRE) V(t) of the geometric Markov renewal processes {St}t∈ℝ+ in (2.5):(2.8)V(t)=∏k=1v(t)D(xk):=D(xv(t))∘D(xv(t)-1)∘⋯∘D(x1).
Using (2.7) we obtain from (2.8)(2.9)V(t)f(s)=∏k=1v(t)D(xk)f(s)=f(s∏k=1v(t)(1+ρ(xk)))=f(St),
where St is defined in (2.5) and S0=s. Let Q(x,A,t) be a semi-Markov kernel for Markov renewal process (xn;θn)n∈Z+, that is, Q(x,A,t)=P(x,A)Gx(t), where P(x,A) is the transition probability of the Markov chain (xn)n∈ℤ+ and Gx(t) is defined by Gx(t):=ℙ(θn+1≤t∣xn=x). Let(2.10)u(t,x):=Ex[V(t)g(x(t))]:=E[V(t)g(x(t))∣x(0)=x]
be the mean value of the semi-Markov random evolution V(t) in (2.9).The following theorem is proved in [1, page 60] and [4, page 38].Theorem 2.3.
The mean valueu(t,x) in (2.10) of the semi-Markov random evolution V(t) given by the solution of the following Markov renewal equation (MRE):
(2.11)u(t,x)-∫0t∫XQ(x,dy,ds)D(y)u(t-s,y)=G¯x(t)g(x),
where G¯x(t)=1-Gx(t),Gx(t):=ℙ(θn+1≤t∣xn=x),  g(x) is a bounded and continuous function on X.


### 2.2. Infinitesimal Operators of the GMRP

Let(2.12)1ρT(x):=ρ(x)T,T>0,(2.13)StT:=S0∏k=1v(tT)(1+ρT(xk))=S0∏k=1v(tT)(1+T-1ρ(xk)).
A detailed information about ρT(x) and StT can be found in Section 4 of [2]. It can be easily shown that(2.14)ln⁡StTS0=∑k=1v(tT)ln⁡(1+ρ(xk)T).
To describe martingale properties of the GMRP (St)t∈ℝ+ in (2.5) we need to find an infinitesimal operator of the process(2.15)η(t):=∑k=1v(t)ln⁡(1+ρ(xk)).
Let γ(t):=t-τv(t) and consider the process (x(t),γ(t)) on X×R+. It is a Markov process with infinitesimal operator(2.16)Q̂f(x,t):=dfdt+gx(t)G¯x(t)∫X[P(x,dy)f(y,0)-f(x,t)],
where gx(t):=dGx(t)/dt,  G¯x(t)=1-Gx(t), where f(x,t)∈C(X×R+). The infinitesimal operator for the process lnS(t) has the form:(2.17)Âf(z,x)=gx(t)G¯x(t)∫XP(x,dy)[f(z+ln⁡(1+ρ(y),x)-f(z,x)],
where z:=lnS0. The process (lnS(t),x(t),γ(t)) is a Markov process on R+×X×R+ with the infinitesimal operator(2.18)L̂f(z,x,t)=Âf(z,x,t)+Q̂f(z,x,t),
where the operators Â and Q̂ are defined in (2.17) and (2.18), respectively. Thus we obtain that the process(2.19)m̂(t):=f(ln⁡S(t),x(t),γ(t))-f(z,x,0)-∫0t(Â+Q̂)f(ln⁡S(u),x(u),γ(u))du
is an ℱ̂t-martingale, where ℱ̂t:=σ(x(s),γ(s);0≤s≤t). If x(t):=xv(t) is a Markov process with kernel(2.20)Q(x,A,t)=P(x,A)(1-e-λ(x)t),
namely, Gx(t)=1-e-λ(x)t, then gx(t)=λ(x)e-λ(x)t,  Ĝx(t)=e-λ(x)t,  gx(t)/Ĝx(t)=λ(x), and the operator Â in (2.17) has the form:(2.21)Âf(z)=λ(x)∫XP(x,dy)[f(z+ln⁡(1+ρ(y)))-f(z)].
The process (lnS(t),x(t)) on R+×X is a Markov process with infinitesimal operator(2.22)L̂f(z,x)=Âf(z,x)+Qf(z,x),
where (2.23)Qf(z,x)=λ(x)∫XP(x,dy)(f(y)-f(x)).
It follows that the process(2.24)m(t):=f(ln⁡S(t),x(t))-f(z,x)-∫0t(Â+Q)f(ln⁡S(u),x(u))du
is an ℱt-martingale, where ℱt:=σ(x(u);0≤u≤t).


### 2.3. Martingale Property of the GMRP

Consider the geometric Markov renewal processes(St)t∈ℝ+(2.25)St=S0∏k=1v(t)(1+ρ(xk)).
For t∈[0,T] let us define(2.26)Lt:=L0∏k=1v(t)h(xk),EL0=1,
where h(x) is a bounded continuous function such that(2.27)∫Xh(y)P(x,dy)=1,∫Xh(y)P(x,dy)p(y)=0.
If ELT=1, then geometric Markov renewal process St in (2.25) is an (ℱt,P*)-martingale, where measure P* is defined as follows:(2.28)dP*dP=LT,Ft:=σ(x(s);0≤s≤t).
In the discrete case we have(2.29)Sn=S0∏k=1n(1+ρ(xk)).
Let Ln:=L0∏k=1nh(xk),EL0=1, where h(x) is defined in (2.27). If ELN=1, then Sn is an (ℱt,P*)-martingale, where dP*/dP=LN, and ℱn:=σ(xk;0≤k≤n).


## 2.1. Jump Semi-Markov Random Evolutions

LetC0(ℝ+) be the space of continuous functions on ℝ+ vanishing at infinity, and let us define a family of bounded contracting operators D(x) on C0(ℝ+) as follows:(2.7)D(x)f(s):=f(s(1+ρ(x)),x∈X,s∈R+.
With these contraction operators D(x) we define the following jump semi-Markov random evolution (JSMRE) V(t) of the geometric Markov renewal processes {St}t∈ℝ+ in (2.5):(2.8)V(t)=∏k=1v(t)D(xk):=D(xv(t))∘D(xv(t)-1)∘⋯∘D(x1).
Using (2.7) we obtain from (2.8)(2.9)V(t)f(s)=∏k=1v(t)D(xk)f(s)=f(s∏k=1v(t)(1+ρ(xk)))=f(St),
where St is defined in (2.5) and S0=s. Let Q(x,A,t) be a semi-Markov kernel for Markov renewal process (xn;θn)n∈Z+, that is, Q(x,A,t)=P(x,A)Gx(t), where P(x,A) is the transition probability of the Markov chain (xn)n∈ℤ+ and Gx(t) is defined by Gx(t):=ℙ(θn+1≤t∣xn=x). Let(2.10)u(t,x):=Ex[V(t)g(x(t))]:=E[V(t)g(x(t))∣x(0)=x]
be the mean value of the semi-Markov random evolution V(t) in (2.9).The following theorem is proved in [1, page 60] and [4, page 38].Theorem 2.3.
The mean valueu(t,x) in (2.10) of the semi-Markov random evolution V(t) given by the solution of the following Markov renewal equation (MRE):
(2.11)u(t,x)-∫0t∫XQ(x,dy,ds)D(y)u(t-s,y)=G¯x(t)g(x),
where G¯x(t)=1-Gx(t),Gx(t):=ℙ(θn+1≤t∣xn=x),  g(x) is a bounded and continuous function on X.


## 2.2. Infinitesimal Operators of the GMRP

Let(2.12)1ρT(x):=ρ(x)T,T>0,(2.13)StT:=S0∏k=1v(tT)(1+ρT(xk))=S0∏k=1v(tT)(1+T-1ρ(xk)).
A detailed information about ρT(x) and StT can be found in Section 4 of [2]. It can be easily shown that(2.14)ln⁡StTS0=∑k=1v(tT)ln⁡(1+ρ(xk)T).
To describe martingale properties of the GMRP (St)t∈ℝ+ in (2.5) we need to find an infinitesimal operator of the process(2.15)η(t):=∑k=1v(t)ln⁡(1+ρ(xk)).
Let γ(t):=t-τv(t) and consider the process (x(t),γ(t)) on X×R+. It is a Markov process with infinitesimal operator(2.16)Q̂f(x,t):=dfdt+gx(t)G¯x(t)∫X[P(x,dy)f(y,0)-f(x,t)],
where gx(t):=dGx(t)/dt,  G¯x(t)=1-Gx(t), where f(x,t)∈C(X×R+). The infinitesimal operator for the process lnS(t) has the form:(2.17)Âf(z,x)=gx(t)G¯x(t)∫XP(x,dy)[f(z+ln⁡(1+ρ(y),x)-f(z,x)],
where z:=lnS0. The process (lnS(t),x(t),γ(t)) is a Markov process on R+×X×R+ with the infinitesimal operator(2.18)L̂f(z,x,t)=Âf(z,x,t)+Q̂f(z,x,t),
where the operators Â and Q̂ are defined in (2.17) and (2.18), respectively. Thus we obtain that the process(2.19)m̂(t):=f(ln⁡S(t),x(t),γ(t))-f(z,x,0)-∫0t(Â+Q̂)f(ln⁡S(u),x(u),γ(u))du
is an ℱ̂t-martingale, where ℱ̂t:=σ(x(s),γ(s);0≤s≤t). If x(t):=xv(t) is a Markov process with kernel(2.20)Q(x,A,t)=P(x,A)(1-e-λ(x)t),
namely, Gx(t)=1-e-λ(x)t, then gx(t)=λ(x)e-λ(x)t,  Ĝx(t)=e-λ(x)t,  gx(t)/Ĝx(t)=λ(x), and the operator Â in (2.17) has the form:(2.21)Âf(z)=λ(x)∫XP(x,dy)[f(z+ln⁡(1+ρ(y)))-f(z)].
The process (lnS(t),x(t)) on R+×X is a Markov process with infinitesimal operator(2.22)L̂f(z,x)=Âf(z,x)+Qf(z,x),
where (2.23)Qf(z,x)=λ(x)∫XP(x,dy)(f(y)-f(x)).
It follows that the process(2.24)m(t):=f(ln⁡S(t),x(t))-f(z,x)-∫0t(Â+Q)f(ln⁡S(u),x(u))du
is an ℱt-martingale, where ℱt:=σ(x(u);0≤u≤t).


## 2.3. Martingale Property of the GMRP

Consider the geometric Markov renewal processes(St)t∈ℝ+(2.25)St=S0∏k=1v(t)(1+ρ(xk)).
For t∈[0,T] let us define(2.26)Lt:=L0∏k=1v(t)h(xk),EL0=1,
where h(x) is a bounded continuous function such that(2.27)∫Xh(y)P(x,dy)=1,∫Xh(y)P(x,dy)p(y)=0.
If ELT=1, then geometric Markov renewal process St in (2.25) is an (ℱt,P*)-martingale, where measure P* is defined as follows:(2.28)dP*dP=LT,Ft:=σ(x(s);0≤s≤t).
In the discrete case we have(2.29)Sn=S0∏k=1n(1+ρ(xk)).
Let Ln:=L0∏k=1nh(xk),EL0=1, where h(x) is defined in (2.27). If ELN=1, then Sn is an (ℱt,P*)-martingale, where dP*/dP=LN, and ℱn:=σ(xk;0≤k≤n).


## 3. Diffusion Approximation of the Geometric Markov Renewal Process (GMRP)

Under an additional balance condition, averaging effect leads to diffusion approximation of the geometric Markov renewal process (GMRP). In fact, we consider the counting processv(t) in (2.5) in the new accelerated scale of time tT2, that is, v≡v(tT2). Due to more rapid changes of states of the system under the balance condition, the fluctuations are described by a diffusion processes.


### 3.1. Ergodic Diffusion Approximation

Let us suppose that balance condition is fulfilled for functionalStT=S0∏k=1v(tT)(1+ρT(xk)):(3.1)ρ̂=∫Xp(dx)∫XP(x,dy)ρ(y)m=0,
where p(x) is ergodic distribution of Markov chain (xk)k∈Z+. Then Ŝ(t)=S0, for all t∈R+. Consider StT in the new scale of time tT2:(3.2)ST(t):=StT2T=S0∏k=1v(tT2)(1+T-1ρ(xk)).
Due to more rapid jumps of v(tT2) the process ST(t) will be fluctuated near the point S0 as T→+∞. By similar arguments similar to (4.3)–(4.5) in [2], we obtain the following expression:(3.3)ln⁡ST(t)S0=T-1∑k=1v(tT2)ρ(xk)-12T-2∑k=1v(tT2)ρ2(xk)+T-2∑k=1v(tT2)r(T-1ρ(xk))ρ2(xk).
Algorithms of ergodic averaging give the limit result for the second term in (3.3) (see [1, page 43] and [4, page 88]):(3.4)lim⁡T→+∞12T-2∑k=1v(tT2)ρ2(xk)=12tρ̂2,
where ρ̂2:=∫Xp(dx)∫XP(x,dy)ρ2(y)/m. Using algorithms of diffusion approximation with respect to the first term in (3.3) we obtain [4, page 88]:(3.5)lim⁡T→+∞T-1∑k=1v(tT2)ρ(xk)=σρw(t),
where σρ2:=∫Xp(dx)[1/2∫XP(x,dy)ρ2(y)+∫XP(x,dy)ρ(y)R0P(x,dy)ρ(y)]/m,  R0 is a potential [3, page 68], of (xn)n∈Z+,  w(t) is a standard Wiener process. The last term in (3.3) goes to zero as T→+∞. Let Ŝ(t) be the limiting process for ST(t) in (3.3) as T→+∞. Taking limit on both sides of (3.3) we obtain(3.6)lim⁡T→+∞ln⁡ST(t)S0=ln⁡Ŝ(t)S0=σρw(t)-12tρ̂2,
where σρ2 and ρ̂2 are defined in (3.4) and (3.5), respectively. From (3.6) we obtain (3.7)Ŝ(t)=S0eσρw(t)-(1/2)tρ̂2=S0e-(1/2)tρ̂2eσρw(t).
Thus, Ŝ(t) satisfies the following stochastic differential equation (SDE):(3.8)dŜ(t)=Ŝ(t)[12(σρ2-ρ̂2)dt+σρdw(t)].In this way we have the following corollary.Corollary 3.1.
The ergodic diffusion GMRP has the form(3.9)Ŝ(t)=S0e-(1/2)tρ̂2eσρw(t),
and it satisfies the following SDE:
(3.10)dŜ(t)Ŝ(t)=12(σρ-ρ̂2)dt+σρdw(t).


### 3.2. Merged Diffusion Approximation

Let us suppose that the balance condition satisfies the following:(3.11)ρ̂(k)=∫Xkpk(dx)∫XkP(x,dy)ρ(y)m(k)=0,
for all k=1,2,…,r where (xn)n∈Z+ is the supporting embedded Markov chain, pk is the stationary density for the ergodic component Xk,  m(k) is defined in [2], and conditions of reducibility of X are fulfilled. Using the algorithms of merged averaging [1, 3, 4] we obtain from the second part of the right hand side in (3.3):(3.12)lim⁡T→+∞12T-1∑k=1v(tT2)ρ2(xk)=12∫0tρ̂2(x̂(s))ds,
where(3.13)ρ̂2(k):=∫Xkpk(dx)∫XkP(x,dy)ρ2(y)m(k)
using the algorithm of merged diffusion approximation that [1, 3, 4] obtain from the first part of the right hand side in (3.3):(3.14)lim⁡T→+∞T-1∑k=1v(tT2)ρ(xk)=∫0tσ̂ρ(x̂(s))dw(s),
where(3.15)σ̂ρ2(k):=∫Xkpk(dx)∫XkP(x,dy)ρ2(y)+∫XkP(x,dy)ρ(y)R0∫XkP(x,dy)ρ(y)m(k).
The third term in (3.3) goes to 0 as T→+∞. In this way, from (3.3) we obtain:(3.16)lim⁡T→+∞ln⁡ST(t)S0=ln⁡S̃(t)S0=∫0tσ̂ρ(x̂(s))dw(s)-12∫0tρ̂2(x̂(s))ds,
where S̃(t) is the limit ST(t) as T→+∞. From (3.16) we obtain(3.17)S̃(t)=S0e-(1/2)∫0tρ̂2(x̂(s))ds+∫0tσ̂ρ(x̂(s))dw(s).
Stochastic differential equation (SDE) for Š(t) has the following form:(3.18)dS̃(t)S̃(t)=12(σ̂ρ2(x̂(t))-ρ̂2(x̂(t)))dt+σ̂ρ(x̂(t))dw(t),
where x̂(t) is a merged Markov process.In this way we have the following corollary.Corollary 3.2.
Merged diffusion GMRP has the form (3.17) and satisfies the SDE (3.18).


### 3.3. Diffusion Approximation under Double Averaging

Let us suppose that the phase spaceX̂={1,2,…,r} of the merged Markov process x̂(t) consists of one ergodic class with stationary distributions (p̂k;k={1,2,…r}). Let us also suppose that the balance condition is fulfilled:(3.19)∑k=1rp̂kρ̂(k)=0.
Then using the algorithms of diffusion approximation under double averaging (see [3, page 188], [1, page 49] and [4, page 93]) we obtain:(3.20)lim⁡T→+∞ln⁡ST(t)S0=ln⁡Š(t)S0=σ̌ρw(t)-12p̌2t,
where(3.21)σ̌ρ2:=∑k=1rp̂kσ̂ρ2(k),p̌2:=∑k=1rp̂kρ̂2(k),
and ρ̂2(k) and σ̂ρ2(k) are defined in (3.13) and (3.15), respectively. Thus, we obtain from (3.20): (3.22)Š(t)=S0e-(1/2)p̌2t+σ̌ρw(t).Corollary 3.3.
The diffusion GMRP under double averaging has the form(3.23)Š(t)=S0e-(1/2)ρ̌2t+σ̌ρw(t),
and satisfies the SDE
(3.24)dŠ(t)Š(t)=12(σ̌ρ2-ρ̌2)dt+σ̌ρdw(t).


## 3.1. Ergodic Diffusion Approximation

Let us suppose that balance condition is fulfilled for functionalStT=S0∏k=1v(tT)(1+ρT(xk)):(3.1)ρ̂=∫Xp(dx)∫XP(x,dy)ρ(y)m=0,
where p(x) is ergodic distribution of Markov chain (xk)k∈Z+. Then Ŝ(t)=S0, for all t∈R+. Consider StT in the new scale of time tT2:(3.2)ST(t):=StT2T=S0∏k=1v(tT2)(1+T-1ρ(xk)).
Due to more rapid jumps of v(tT2) the process ST(t) will be fluctuated near the point S0 as T→+∞. By similar arguments similar to (4.3)–(4.5) in [2], we obtain the following expression:(3.3)ln⁡ST(t)S0=T-1∑k=1v(tT2)ρ(xk)-12T-2∑k=1v(tT2)ρ2(xk)+T-2∑k=1v(tT2)r(T-1ρ(xk))ρ2(xk).
Algorithms of ergodic averaging give the limit result for the second term in (3.3) (see [1, page 43] and [4, page 88]):(3.4)lim⁡T→+∞12T-2∑k=1v(tT2)ρ2(xk)=12tρ̂2,
where ρ̂2:=∫Xp(dx)∫XP(x,dy)ρ2(y)/m. Using algorithms of diffusion approximation with respect to the first term in (3.3) we obtain [4, page 88]:(3.5)lim⁡T→+∞T-1∑k=1v(tT2)ρ(xk)=σρw(t),
where σρ2:=∫Xp(dx)[1/2∫XP(x,dy)ρ2(y)+∫XP(x,dy)ρ(y)R0P(x,dy)ρ(y)]/m,  R0 is a potential [3, page 68], of (xn)n∈Z+,  w(t) is a standard Wiener process. The last term in (3.3) goes to zero as T→+∞. Let Ŝ(t) be the limiting process for ST(t) in (3.3) as T→+∞. Taking limit on both sides of (3.3) we obtain(3.6)lim⁡T→+∞ln⁡ST(t)S0=ln⁡Ŝ(t)S0=σρw(t)-12tρ̂2,
where σρ2 and ρ̂2 are defined in (3.4) and (3.5), respectively. From (3.6) we obtain (3.7)Ŝ(t)=S0eσρw(t)-(1/2)tρ̂2=S0e-(1/2)tρ̂2eσρw(t).
Thus, Ŝ(t) satisfies the following stochastic differential equation (SDE):(3.8)dŜ(t)=Ŝ(t)[12(σρ2-ρ̂2)dt+σρdw(t)].In this way we have the following corollary.Corollary 3.1.
The ergodic diffusion GMRP has the form(3.9)Ŝ(t)=S0e-(1/2)tρ̂2eσρw(t),
and it satisfies the following SDE:
(3.10)dŜ(t)Ŝ(t)=12(σρ-ρ̂2)dt+σρdw(t).


## 3.2. Merged Diffusion Approximation

Let us suppose that the balance condition satisfies the following:(3.11)ρ̂(k)=∫Xkpk(dx)∫XkP(x,dy)ρ(y)m(k)=0,
for all k=1,2,…,r where (xn)n∈Z+ is the supporting embedded Markov chain, pk is the stationary density for the ergodic component Xk,  m(k) is defined in [2], and conditions of reducibility of X are fulfilled. Using the algorithms of merged averaging [1, 3, 4] we obtain from the second part of the right hand side in (3.3):(3.12)lim⁡T→+∞12T-1∑k=1v(tT2)ρ2(xk)=12∫0tρ̂2(x̂(s))ds,
where(3.13)ρ̂2(k):=∫Xkpk(dx)∫XkP(x,dy)ρ2(y)m(k)
using the algorithm of merged diffusion approximation that [1, 3, 4] obtain from the first part of the right hand side in (3.3):(3.14)lim⁡T→+∞T-1∑k=1v(tT2)ρ(xk)=∫0tσ̂ρ(x̂(s))dw(s),
where(3.15)σ̂ρ2(k):=∫Xkpk(dx)∫XkP(x,dy)ρ2(y)+∫XkP(x,dy)ρ(y)R0∫XkP(x,dy)ρ(y)m(k).
The third term in (3.3) goes to 0 as T→+∞. In this way, from (3.3) we obtain:(3.16)lim⁡T→+∞ln⁡ST(t)S0=ln⁡S̃(t)S0=∫0tσ̂ρ(x̂(s))dw(s)-12∫0tρ̂2(x̂(s))ds,
where S̃(t) is the limit ST(t) as T→+∞. From (3.16) we obtain(3.17)S̃(t)=S0e-(1/2)∫0tρ̂2(x̂(s))ds+∫0tσ̂ρ(x̂(s))dw(s).
Stochastic differential equation (SDE) for Š(t) has the following form:(3.18)dS̃(t)S̃(t)=12(σ̂ρ2(x̂(t))-ρ̂2(x̂(t)))dt+σ̂ρ(x̂(t))dw(t),
where x̂(t) is a merged Markov process.In this way we have the following corollary.Corollary 3.2.
Merged diffusion GMRP has the form (3.17) and satisfies the SDE (3.18).


## 3.3. Diffusion Approximation under Double Averaging

Let us suppose that the phase spaceX̂={1,2,…,r} of the merged Markov process x̂(t) consists of one ergodic class with stationary distributions (p̂k;k={1,2,…r}). Let us also suppose that the balance condition is fulfilled:(3.19)∑k=1rp̂kρ̂(k)=0.
Then using the algorithms of diffusion approximation under double averaging (see [3, page 188], [1, page 49] and [4, page 93]) we obtain:(3.20)lim⁡T→+∞ln⁡ST(t)S0=ln⁡Š(t)S0=σ̌ρw(t)-12p̌2t,
where(3.21)σ̌ρ2:=∑k=1rp̂kσ̂ρ2(k),p̌2:=∑k=1rp̂kρ̂2(k),
and ρ̂2(k) and σ̂ρ2(k) are defined in (3.13) and (3.15), respectively. Thus, we obtain from (3.20): (3.22)Š(t)=S0e-(1/2)p̌2t+σ̌ρw(t).Corollary 3.3.
The diffusion GMRP under double averaging has the form(3.23)Š(t)=S0e-(1/2)ρ̌2t+σ̌ρw(t),
and satisfies the SDE
(3.24)dŠ(t)Š(t)=12(σ̌ρ2-ρ̌2)dt+σ̌ρdw(t).


## 4. Proofs

In this section we present proofs of results in Section3. All the above-mentioned results are obtained from the general results for semi-Markov random evolutions [3, 4] in series scheme. The main steps of proof are (1) weak convergence of StT in Skorokhod space DR[0,+∞) [7, page 148]; (2) solution of martingale problem for the limit process Ŝ(t); (3) characterization of the limit measure for the limit process Ŝ(t); (4) uniqueness of solution of martingale problem. We also give here the rate of convergence in the diffusion approximation scheme.


### 4.1. Diffusion Approximation (DA)

Let(4.1)GtT:=T-1∑k=0v(tT2)ρ(xk),GnT:=GτnT-1T,G0T=ln⁡s,
and the balance condition is satisfied:(4.2)ρ̂:=∫Xp(dx)∫XP(x,dy)ρ(y)=0.
Let us define the functions(4.3)ϕT(s,x):=f(s)+T-1ϕf1(s,x)+T-2ϕf2(s,x),
where ϕf1 and ϕf2 are defined as follows:(4.4)(P-I)ϕf1(s,x)=ρ(x)f(s),(P-I)ϕf2(s,x)=[-A(x)+Â]f(s),
where(4.5)Â:=∫Xp(dx)A(x),
and A(x):=[ρ2(x)/2+ρ(x)(R0-I)ρ(x)]d2/ds2. From the balance condition (4.2) and equality Π(Â-A(x))=0 it follows that both equations in (4.3) simultaneously solvable and the solutions ϕfi(s,x) are bounded functions, i=1,2.We note that(4.6)f(Sn+1T)-f(GnT)=1Tρ(xn)df(xn)ds
and define(4.7)ϕT(s,x):=f(s)+T-1ϕf1(s,x)+T-2ϕf2(s,x),
where ϕf1(s,x) and ϕf2(s,x) are defined in (4.4) and (4.5), respectively. We note, that Gn+1T-GnT=T-1ρ(xn).


### 4.2. Martingale Problem for the Limiting ProblemG0(t) in DA

Let us introduce the family of functions:(4.8)ψT(s,t):=ϕT(G[tT2]T,x[tT2])-ϕT(G[st2]T,x[sT2])-∑j=[sT2][tT2]-1E[ϕT(GjT,xj+1)-ϕT(GjT,xj)∣Fj],
where ϕT are defined in (4.7) and GjT is defined by (4.9)Gτn/TT=1T∑k=0nρ(xk).
Functions ψT(s,t) are ℱ[tT2]-martingale by t. Taking into account the expression (4.6) and (4.7), we find the following expression:(4.10)ψT(s,t)=f(G[tT2]T)-f(G[sT2]T)+ϵ[ϕf1(G[tT2]T,x[tT2])-ϕf1(G[st2]T,x[sT2])]+ϵ2[ϕf2(G[tT2]T,x[tT2])-ϕf2(G[sT2]T,x[sT2])]-T-1∑j=[sT2][tT2]-1{ρ(xj)df(GjT)dg+E(ϕf1(GjT,xj+1)-ϕf2(GjT,xj)∣Fj)}-T-2∑j=[sT2][tT2]-1{2-1ρ2(xj)df(GjT)dg+ρ(xj)E(dϕf1(GjT,xj+1)dg∣Fj)+E[ϕf2(GjT,xj+1)-ϕf2(GjT,xj)∣Fj]df(GjT)dg}+o(T-2)=f(G[tT2]T)-f(G[sT2]T)+[ϕf1(G[tT2]T,x[tT2])-ϕf1(G[sT2]T,x[sT2])]+T-2[ϕf2(G[tT2]T,x[tT2])-ϕf2(G[sT2]T,x[sT2])]-T-2∑j=[sT2][tT2]-1Âf(GjT)+O(T-2),
where O(T-2) is the sum of terms with T-2nd order. Since ψT(0,t) is ℱ[tT2]-martingale with respect to measure QT, generated by process GT(t) in (4.1), then for every scalar linear continuous functional η0s we have from (4.8)-(4.10):(4.11)0=ET[(ψT(s,t)η0s]=ET[(f(G[tT2]T)-f(G[sT2]T)-T-2∑j=[sT2][tT2]-1Âf(GjT))η0s]-T-1ET[(ϕf1(G[tT2]T,x[tT2])-ϕf1(G[sT2]T,x[sT2]))η0s]-T-2ET[(ϕf2(G[tT2]T,x[tT2])-ϕf2(G[sT2]T,x[sT2]))η0s]-O(T-2),
where ET is a mean value by measure QT. If the process G[tT2]T converges weakly to some process G0(t) as T→+∞, then from (4.11) we obtain(4.12)0=ET[(f(G0(t))-f(G0(s))-∫stÂf(G0(u))du],
that is, the process (4.13)f(G0(t))-f(G0(s))-∫stÂf(G0(u))du
is a continuous QT-martingale. Since Â is the second order differential operator and coefficient σ12 is positively defined, where(4.14)σ12:=∫Xπ(dx)[ρ2(x)2+ρ(x)R0ρ(x)],
then the process G0(t) is a Wiener process with variance σ12 in (4.14): G0(t)=σw(t). Taking into account the renewal theorem for v(t), namely, T-1v(tT2)→T→+∞t/m, and the following representation (4.15)GT(t)=T-1∑k=0v(tT2)ρ(xk)=T-1∑k=0[tT2]ρ(xk)+T-1∑k=[tT2]+1v(tT2)ρ(xk)
we obtain, replacing [tT2] by v(tT2), that process GT(t) converges weakly to the process Ĝ0(t) as T→+∞, which is the solution of such martingale problem: (4.16)f(Ĝ0(t))-f(Ĝ0(s))-∫stÂ0f(Ĝ0(u))du
is a continuous QT-martingale, where Â0:=Â/m, and Â is defined in (4.5)-(4.5).


### 4.3. Weak Convergence of the ProcessesGT(t) in DA

From the representation of the processGT(t) it follows that(4.17)ΔT(s,t):=|GT(t)-GT(s)|=|T-1∑k=v(sT2)+1v(tT2)ρ(xk)|≤T-1sup⁡xρ(x)|v(tT2)-v(sT2)-1|.
This representation gives the following estimation:(4.18)|ΔT(t1,t2)||ΔT(t2,t3)|≤T-2(sup⁡xρ(x))2|v(t3T2)-v(t1T2)|2.
Taking into account the same reasonings as in [2] we obtain the weak convergence of the processes GT(t) in DA.


### 4.4. Characterization of the Limiting MeasureQ for QT as T→+∞ in DA

From Section4.3 (see also Section 4.1.4 of [2]) it follows that there exists a sequence Tn such that measures QTn converge weakly to some measure Q on DR[0,+∞) as T→+∞, where DR[0,+∞) is the Skorokhod space [7, page 148]. This measure is the solution of such martingale problem: the following process(4.19)m(s,t):=f(Ĝ0(t))-f(Ĝ0(s))-∫stÂ0f(Ĝ0(u))du
is a Q-martingale for all f(g)∈C2(R) and(4.20)Em(s,t)η0s=0,
for scalar continuous bounded functional η0s,  E is a mean value by measure Q. From (4.19) it follows that ETmT(s,t)η0s=0, and it is necessary to show that the limiting passing in (4.1) goes to the process in (3.12) as T→+∞. From equality (4.11) we find that limTn→+∞ETnm(s,t)η0s=Em(s,t)η0s. Moreover, from the following expression (4.21)|ETm(s,t)η0s-Em(s,t)η0s|≤|(ET-E)m(s,t)η0s|+ET|m(s,t)-mT(s,t)||mTη0s|⟶T→+∞0,
we obtain that there exists the measure Q on DR[0,+∞) which solves the martingale problem for the operator Â0 (or, equivalently, for the process Ĝ0(t) in the form (4.12)). Uniqueness of the solution of the martingale problem follows from the fact that operator Â0 generates the unique semigroup with respects to the Wiener process with variance σ12 in (4.14). As long as the semigroup is unique then the limit process Ĝ0(t) is unique. See [3, Chapter 1].


### 4.5. Calculation of the Quadratic Variation for GMRP

IfGnT=GT-1τnT, the sequence(4.22)mnT:=GnT-G0T-∑k=0n-1E[Gk+1T-GkT∣Fk],G0T=g,
is ℱn-martingale, where ℱn:=σ{xk,θk;0≤k≤n}. From the definition it follows that the characteristic 〈mnT〉 of the martingale mnT has the form(4.23)〈mnT〉=∑k=0n-1E[(mk+1T-mkT)2∣Fk].
To calculate 〈mnT〉 let us represent mnT in (4.22) in the form of martingale-difference: (4.24)mnT=∑k=0n-1[Gk+1T-E(Gk+1T∣Fk)].
From representation (4.25)Gn+1T-GnT=1Tρ(xn)
it follows that E(Gk+1T∣ℱk)=GkT+T-1ρ(xk), that is why(4.26)Gk+1T-E(Gk+1T∣Fk)=T-1(ρ(xk)-Pρ(xk)).
Since from (4.22) it follows that(4.27)mk+1T-mkT=Gk+1T-E(Gk+1T∣Fk)=T-1(ρ(xk)-Pρ(xk)),
then substituting (4.27) in (4.23) we obtain(4.28)〈mnT〉=T-2∑k=0n-1[(I-P)ρ(xk)]2.
In an averaging scheme (see [2]) for GMRP in the scale of time tT we obtain that 〈m[tT]T〉 goes to zero as T→+∞ in probability, which follows from (4.27): (4.29)〈m[tT]T〉=T-2∑k=0[tT]-1[(I-P)ρ(xk)]2⟶0asT⟶+∞
for all t∈R+. In the diffusion approximation scheme for GMRP in scale of time tT2 from (4.27) we obtain that characteristic 〈m[tT2]T〉 does not go to zero as T→+∞ since (4.30)〈m[tT2]T〉=T-2∑k=0[tT2]-1[(I-P)ρ(xk)]2⟶tσ12,
where σ12:=∫Xπ(dx)[(I-P)ρ(x)]2.


### 4.6. Rates of Convergence for GMRP

Consider the representation (4.22) for martingale mnT. It follows that(4.31)GnT=g+mnT+∑k=0n-1E[Gk+1T-GkT∣Fk].
In diffusion approximation scheme for GMRP the limit for the process G[tT2]T as T→+∞ will be diffusion process Ŝ(t) (see (3.10)). If m0(t) is the limiting martingale for m[tT2]T in (4.22) as T→+∞, then from (4.31) and (3.10) we obtain
(4.32)E[G[tT2]T-Ŝ(t)]=E[m[tT2]T-m0(t)]+T-1∑k=0[tT2]-1ρ(xk)-Ŝ(t).
Since E[m[tT2]T-m0(t)]=0, (because m[tT2]T and m0(t) are zero-mean martingales) then from (4.32) we obtain:(4.33)|E[G[tT2]T-Ŝ(t)]|≤T-1|∑k=0[tT2]-1ρ(xk)-Ŝ(t)T|.
Taking into account the balance condition ∫Xπ(dx)ρ(x)=0 and the central limit theorem for a Markov chain [4, page 98], we obtain(4.34)|∑k=0[tT2]-1ρ(xk)-Ŝ(t)T|=C1(t0),
where C1(t0) is a constant depending on t0,t∈[0,t0]. From (4.33), (4.2), and (4.32) we obtain:(4.35)|E[G[tT2]T-Ŝ(t)]|≤T-1C1(t0).
Thus, the rates of convergence in diffusion scheme has the order T-1.


## 4.1. Diffusion Approximation (DA)

Let(4.1)GtT:=T-1∑k=0v(tT2)ρ(xk),GnT:=GτnT-1T,G0T=ln⁡s,
and the balance condition is satisfied:(4.2)ρ̂:=∫Xp(dx)∫XP(x,dy)ρ(y)=0.
Let us define the functions(4.3)ϕT(s,x):=f(s)+T-1ϕf1(s,x)+T-2ϕf2(s,x),
where ϕf1 and ϕf2 are defined as follows:(4.4)(P-I)ϕf1(s,x)=ρ(x)f(s),(P-I)ϕf2(s,x)=[-A(x)+Â]f(s),
where(4.5)Â:=∫Xp(dx)A(x),
and A(x):=[ρ2(x)/2+ρ(x)(R0-I)ρ(x)]d2/ds2. From the balance condition (4.2) and equality Π(Â-A(x))=0 it follows that both equations in (4.3) simultaneously solvable and the solutions ϕfi(s,x) are bounded functions, i=1,2.We note that(4.6)f(Sn+1T)-f(GnT)=1Tρ(xn)df(xn)ds
and define(4.7)ϕT(s,x):=f(s)+T-1ϕf1(s,x)+T-2ϕf2(s,x),
where ϕf1(s,x) and ϕf2(s,x) are defined in (4.4) and (4.5), respectively. We note, that Gn+1T-GnT=T-1ρ(xn).


## 4.2. Martingale Problem for the Limiting ProblemG0(t) in DA

Let us introduce the family of functions:(4.8)ψT(s,t):=ϕT(G[tT2]T,x[tT2])-ϕT(G[st2]T,x[sT2])-∑j=[sT2][tT2]-1E[ϕT(GjT,xj+1)-ϕT(GjT,xj)∣Fj],
where ϕT are defined in (4.7) and GjT is defined by (4.9)Gτn/TT=1T∑k=0nρ(xk).
Functions ψT(s,t) are ℱ[tT2]-martingale by t. Taking into account the expression (4.6) and (4.7), we find the following expression:(4.10)ψT(s,t)=f(G[tT2]T)-f(G[sT2]T)+ϵ[ϕf1(G[tT2]T,x[tT2])-ϕf1(G[st2]T,x[sT2])]+ϵ2[ϕf2(G[tT2]T,x[tT2])-ϕf2(G[sT2]T,x[sT2])]-T-1∑j=[sT2][tT2]-1{ρ(xj)df(GjT)dg+E(ϕf1(GjT,xj+1)-ϕf2(GjT,xj)∣Fj)}-T-2∑j=[sT2][tT2]-1{2-1ρ2(xj)df(GjT)dg+ρ(xj)E(dϕf1(GjT,xj+1)dg∣Fj)+E[ϕf2(GjT,xj+1)-ϕf2(GjT,xj)∣Fj]df(GjT)dg}+o(T-2)=f(G[tT2]T)-f(G[sT2]T)+[ϕf1(G[tT2]T,x[tT2])-ϕf1(G[sT2]T,x[sT2])]+T-2[ϕf2(G[tT2]T,x[tT2])-ϕf2(G[sT2]T,x[sT2])]-T-2∑j=[sT2][tT2]-1Âf(GjT)+O(T-2),
where O(T-2) is the sum of terms with T-2nd order. Since ψT(0,t) is ℱ[tT2]-martingale with respect to measure QT, generated by process GT(t) in (4.1), then for every scalar linear continuous functional η0s we have from (4.8)-(4.10):(4.11)0=ET[(ψT(s,t)η0s]=ET[(f(G[tT2]T)-f(G[sT2]T)-T-2∑j=[sT2][tT2]-1Âf(GjT))η0s]-T-1ET[(ϕf1(G[tT2]T,x[tT2])-ϕf1(G[sT2]T,x[sT2]))η0s]-T-2ET[(ϕf2(G[tT2]T,x[tT2])-ϕf2(G[sT2]T,x[sT2]))η0s]-O(T-2),
where ET is a mean value by measure QT. If the process G[tT2]T converges weakly to some process G0(t) as T→+∞, then from (4.11) we obtain(4.12)0=ET[(f(G0(t))-f(G0(s))-∫stÂf(G0(u))du],
that is, the process (4.13)f(G0(t))-f(G0(s))-∫stÂf(G0(u))du
is a continuous QT-martingale. Since Â is the second order differential operator and coefficient σ12 is positively defined, where(4.14)σ12:=∫Xπ(dx)[ρ2(x)2+ρ(x)R0ρ(x)],
then the process G0(t) is a Wiener process with variance σ12 in (4.14): G0(t)=σw(t). Taking into account the renewal theorem for v(t), namely, T-1v(tT2)→T→+∞t/m, and the following representation (4.15)GT(t)=T-1∑k=0v(tT2)ρ(xk)=T-1∑k=0[tT2]ρ(xk)+T-1∑k=[tT2]+1v(tT2)ρ(xk)
we obtain, replacing [tT2] by v(tT2), that process GT(t) converges weakly to the process Ĝ0(t) as T→+∞, which is the solution of such martingale problem: (4.16)f(Ĝ0(t))-f(Ĝ0(s))-∫stÂ0f(Ĝ0(u))du
is a continuous QT-martingale, where Â0:=Â/m, and Â is defined in (4.5)-(4.5).


## 4.3. Weak Convergence of the ProcessesGT(t) in DA

From the representation of the processGT(t) it follows that(4.17)ΔT(s,t):=|GT(t)-GT(s)|=|T-1∑k=v(sT2)+1v(tT2)ρ(xk)|≤T-1sup⁡xρ(x)|v(tT2)-v(sT2)-1|.
This representation gives the following estimation:(4.18)|ΔT(t1,t2)||ΔT(t2,t3)|≤T-2(sup⁡xρ(x))2|v(t3T2)-v(t1T2)|2.
Taking into account the same reasonings as in [2] we obtain the weak convergence of the processes GT(t) in DA.


## 4.4. Characterization of the Limiting MeasureQ for QT as T→+∞ in DA

From Section4.3 (see also Section 4.1.4 of [2]) it follows that there exists a sequence Tn such that measures QTn converge weakly to some measure Q on DR[0,+∞) as T→+∞, where DR[0,+∞) is the Skorokhod space [7, page 148]. This measure is the solution of such martingale problem: the following process(4.19)m(s,t):=f(Ĝ0(t))-f(Ĝ0(s))-∫stÂ0f(Ĝ0(u))du
is a Q-martingale for all f(g)∈C2(R) and(4.20)Em(s,t)η0s=0,
for scalar continuous bounded functional η0s,  E is a mean value by measure Q. From (4.19) it follows that ETmT(s,t)η0s=0, and it is necessary to show that the limiting passing in (4.1) goes to the process in (3.12) as T→+∞. From equality (4.11) we find that limTn→+∞ETnm(s,t)η0s=Em(s,t)η0s. Moreover, from the following expression (4.21)|ETm(s,t)η0s-Em(s,t)η0s|≤|(ET-E)m(s,t)η0s|+ET|m(s,t)-mT(s,t)||mTη0s|⟶T→+∞0,
we obtain that there exists the measure Q on DR[0,+∞) which solves the martingale problem for the operator Â0 (or, equivalently, for the process Ĝ0(t) in the form (4.12)). Uniqueness of the solution of the martingale problem follows from the fact that operator Â0 generates the unique semigroup with respects to the Wiener process with variance σ12 in (4.14). As long as the semigroup is unique then the limit process Ĝ0(t) is unique. See [3, Chapter 1].


## 4.5. Calculation of the Quadratic Variation for GMRP

IfGnT=GT-1τnT, the sequence(4.22)mnT:=GnT-G0T-∑k=0n-1E[Gk+1T-GkT∣Fk],G0T=g,
is ℱn-martingale, where ℱn:=σ{xk,θk;0≤k≤n}. From the definition it follows that the characteristic 〈mnT〉 of the martingale mnT has the form(4.23)〈mnT〉=∑k=0n-1E[(mk+1T-mkT)2∣Fk].
To calculate 〈mnT〉 let us represent mnT in (4.22) in the form of martingale-difference: (4.24)mnT=∑k=0n-1[Gk+1T-E(Gk+1T∣Fk)].
From representation (4.25)Gn+1T-GnT=1Tρ(xn)
it follows that E(Gk+1T∣ℱk)=GkT+T-1ρ(xk), that is why(4.26)Gk+1T-E(Gk+1T∣Fk)=T-1(ρ(xk)-Pρ(xk)).
Since from (4.22) it follows that(4.27)mk+1T-mkT=Gk+1T-E(Gk+1T∣Fk)=T-1(ρ(xk)-Pρ(xk)),
then substituting (4.27) in (4.23) we obtain(4.28)〈mnT〉=T-2∑k=0n-1[(I-P)ρ(xk)]2.
In an averaging scheme (see [2]) for GMRP in the scale of time tT we obtain that 〈m[tT]T〉 goes to zero as T→+∞ in probability, which follows from (4.27): (4.29)〈m[tT]T〉=T-2∑k=0[tT]-1[(I-P)ρ(xk)]2⟶0asT⟶+∞
for all t∈R+. In the diffusion approximation scheme for GMRP in scale of time tT2 from (4.27) we obtain that characteristic 〈m[tT2]T〉 does not go to zero as T→+∞ since (4.30)〈m[tT2]T〉=T-2∑k=0[tT2]-1[(I-P)ρ(xk)]2⟶tσ12,
where σ12:=∫Xπ(dx)[(I-P)ρ(x)]2.


## 4.6. Rates of Convergence for GMRP

Consider the representation (4.22) for martingale mnT. It follows that(4.31)GnT=g+mnT+∑k=0n-1E[Gk+1T-GkT∣Fk].
In diffusion approximation scheme for GMRP the limit for the process G[tT2]T as T→+∞ will be diffusion process Ŝ(t) (see (3.10)). If m0(t) is the limiting martingale for m[tT2]T in (4.22) as T→+∞, then from (4.31) and (3.10) we obtain
(4.32)E[G[tT2]T-Ŝ(t)]=E[m[tT2]T-m0(t)]+T-1∑k=0[tT2]-1ρ(xk)-Ŝ(t).
Since E[m[tT2]T-m0(t)]=0, (because m[tT2]T and m0(t) are zero-mean martingales) then from (4.32) we obtain:(4.33)|E[G[tT2]T-Ŝ(t)]|≤T-1|∑k=0[tT2]-1ρ(xk)-Ŝ(t)T|.
Taking into account the balance condition ∫Xπ(dx)ρ(x)=0 and the central limit theorem for a Markov chain [4, page 98], we obtain(4.34)|∑k=0[tT2]-1ρ(xk)-Ŝ(t)T|=C1(t0),
where C1(t0) is a constant depending on t0,t∈[0,t0]. From (4.33), (4.2), and (4.32) we obtain:(4.35)|E[G[tT2]T-Ŝ(t)]|≤T-1C1(t0).
Thus, the rates of convergence in diffusion scheme has the order T-1.


## 5. Merged Diffusion Geometric Markov Renewal Process in the Case of Two Ergodic Classes

### 5.1. Two Ergodic Classes

LetP(x,A):=ℙ{xn+1∈A∣xn=x} be the transition probabilities of supporting embedded reducible Markov chain {xn}n≥0 in the phase space X. Let us have two ergodic classes X0 and X1 of the phase space such that:(5.1)X=X0∪X1,X0∩X1=∅.
Let {X̂={0,1},𝒱} be the measurable merged phase space. A stochastic kernel P0(x,A) is consistent with the splitting (5.1) in the following way: (5.2)P0(x,Xk)=1k:={1,x∈Xk,0,x∉Xk,k=0,1.
Let the supporting embedded Markov chain (xn)n∈Z+ with the transition probabilities P0(x,A) be uniformly ergodic in each class Xk,k=0,1 and have a stationary distribution πk(dx) in the classes Xk,  k=0,1: (5.3)πk(A)=∫Xkπk(dx)P0(x,A),A⊂Xk,k=0,1.
Let the stationary escape probabilities of the embedded Markov chain (xn)n∈Z+ with transition probabilities P(x,A):=ℙ{xn+1∈A∣xn=x} be positive and sufficiently small, that is, (5.4)qk(A)=∫Xkπk(dx)P(x,X∖Xk)>0,k=0,1.
Let the stationary sojourn time in the classes of states be uniformly bounded, namely,(5.5)0≤C1≤mk:=∫Xkπk(dx)m(x)≤C2,k=0,1,
where (5.6)m(x):=∫0∞G¯x(t)dt.


### 5.2. Algorithms of Phase Averaging with Two Ergodic Classes

The merged Markov chain(x̂n)n∈Z+ in merged phase space X̂ is given by matrix of transition probabilities(5.7)P̂=(p̂kr)k,r=0,1;p̂01=1-p̂11=∫X1π1(dx)P(x,X0)=1-∫X1π1(dx)P(x,X1);p̂01=1-p̂00=∫X0π0(dx)P(x,X1)=1-∫X0π0(dx)P(x,X0).
As p̂kr≠0, k=0,1, then x̂n has virtual transitions. Intensities Λ̂k of sojourn times θ̂k, k=0,1, of the merged MRP are calculated as follows:(5.8)Λ̂k=1mk,mk=∫Xkπk(dx)m(x),k=0,1.
And, finally, the merged MRP (x̂n,θ̂)n∈Z+ in the merged phase space X̂ is given by the stochastic matrix(5.9)Q̂(t)=(Q̂kr)k,r=0,1:=p̂kr(1-e-Λ̂kt),k,r=0,1.
Hence, the initial semi-Markov system is merged to a Markov system with two classes.


### 5.3. Merged Diffusion Approximation in the Case of Two Ergodic Classes

The merged diffusion GMRP in the case of two ergodic classes has the form:(5.10)S̃(t)=S0e-(1/2)∫0tρ̂2(x̂(s))ds+∫0tσ̂ρ(x̂(s))dw(s)
which satisfies the stochastic differential equation (SDE):(5.11)dS̃(t)S̃(t)=12(σ̂ρ2(x̂(t))-ρ̂2(x̂(t)))dt+σ̂ρ(x̂(t))dw(t),
where (5.12)ρ̂2(1):=∫X1p1(dx)∫X1P(x,dy)ρ2(y)m(1),ρ̂2(0):=∫X0p0(dx)∫X0P(x,dy)ρ2(y)m(0),σ̂ρ2(1):=∫X1p1(dx)∫X1P(x,dy)ρ2(y)+∫X1P(x,dy)ρ(y)R0∫X1P(x,dy)ρ(y)m(1),σ̂ρ2(0):=∫X0p0(dx)∫X0P(x,dy)ρ2(y)+∫X0P(x,dy)ρ(y)R0∫X0P(x,dy)ρ(y)m(0),x̂(t) is a merged Markov process in X̂={0,1} with stochastic matrix Q̂(t) in (5.9).


## 5.1. Two Ergodic Classes

LetP(x,A):=ℙ{xn+1∈A∣xn=x} be the transition probabilities of supporting embedded reducible Markov chain {xn}n≥0 in the phase space X. Let us have two ergodic classes X0 and X1 of the phase space such that:(5.1)X=X0∪X1,X0∩X1=∅.
Let {X̂={0,1},𝒱} be the measurable merged phase space. A stochastic kernel P0(x,A) is consistent with the splitting (5.1) in the following way: (5.2)P0(x,Xk)=1k:={1,x∈Xk,0,x∉Xk,k=0,1.
Let the supporting embedded Markov chain (xn)n∈Z+ with the transition probabilities P0(x,A) be uniformly ergodic in each class Xk,k=0,1 and have a stationary distribution πk(dx) in the classes Xk,  k=0,1: (5.3)πk(A)=∫Xkπk(dx)P0(x,A),A⊂Xk,k=0,1.
Let the stationary escape probabilities of the embedded Markov chain (xn)n∈Z+ with transition probabilities P(x,A):=ℙ{xn+1∈A∣xn=x} be positive and sufficiently small, that is, (5.4)qk(A)=∫Xkπk(dx)P(x,X∖Xk)>0,k=0,1.
Let the stationary sojourn time in the classes of states be uniformly bounded, namely,(5.5)0≤C1≤mk:=∫Xkπk(dx)m(x)≤C2,k=0,1,
where (5.6)m(x):=∫0∞G¯x(t)dt.


## 5.2. Algorithms of Phase Averaging with Two Ergodic Classes

The merged Markov chain(x̂n)n∈Z+ in merged phase space X̂ is given by matrix of transition probabilities(5.7)P̂=(p̂kr)k,r=0,1;p̂01=1-p̂11=∫X1π1(dx)P(x,X0)=1-∫X1π1(dx)P(x,X1);p̂01=1-p̂00=∫X0π0(dx)P(x,X1)=1-∫X0π0(dx)P(x,X0).
As p̂kr≠0, k=0,1, then x̂n has virtual transitions. Intensities Λ̂k of sojourn times θ̂k, k=0,1, of the merged MRP are calculated as follows:(5.8)Λ̂k=1mk,mk=∫Xkπk(dx)m(x),k=0,1.
And, finally, the merged MRP (x̂n,θ̂)n∈Z+ in the merged phase space X̂ is given by the stochastic matrix(5.9)Q̂(t)=(Q̂kr)k,r=0,1:=p̂kr(1-e-Λ̂kt),k,r=0,1.
Hence, the initial semi-Markov system is merged to a Markov system with two classes.


## 5.3. Merged Diffusion Approximation in the Case of Two Ergodic Classes

The merged diffusion GMRP in the case of two ergodic classes has the form:(5.10)S̃(t)=S0e-(1/2)∫0tρ̂2(x̂(s))ds+∫0tσ̂ρ(x̂(s))dw(s)
which satisfies the stochastic differential equation (SDE):(5.11)dS̃(t)S̃(t)=12(σ̂ρ2(x̂(t))-ρ̂2(x̂(t)))dt+σ̂ρ(x̂(t))dw(t),
where (5.12)ρ̂2(1):=∫X1p1(dx)∫X1P(x,dy)ρ2(y)m(1),ρ̂2(0):=∫X0p0(dx)∫X0P(x,dy)ρ2(y)m(0),σ̂ρ2(1):=∫X1p1(dx)∫X1P(x,dy)ρ2(y)+∫X1P(x,dy)ρ(y)R0∫X1P(x,dy)ρ(y)m(1),σ̂ρ2(0):=∫X0p0(dx)∫X0P(x,dy)ρ2(y)+∫X0P(x,dy)ρ(y)R0∫X0P(x,dy)ρ(y)m(0),x̂(t) is a merged Markov process in X̂={0,1} with stochastic matrix Q̂(t) in (5.9).


## 6. European Call Option Pricing Formulas for Diffusion GMRP

### 6.1. Ergodic Geometric Markov Renewal Process

As we have seen in Section3, an ergodic diffusion GMRP Ŝ(t) satisfies the following SDE (see (3.10)):
(6.1)dŜ(t)Ŝ(t)=12(σρ-ρ̂2)dt+σρdw(t),
where(6.2)ρ̂2=∫Xp(dx)∫XP(x,dy)ρ2(y)m,(6.3)σρ2=∫Xp(dx)[12∫XP(x,dy)ρ2(y)+∫XP(x,dy)ρ(y)R0P(x,dy)ρ(y)m.
The risk-neutral measure P* for the process in (6.1) is:(6.4)dP*P=exp⁡{-θt-12θ2w(t)},
where (6.5)θ=((1/2)(σρ-ρ̂2)-r)σρ.
Under P*, the process e-rtŜt is a martingale and the process w*(t)=w(t)+θt is a Brownian motion. In this way, in the risk-neutral world, the process Ŝt has the following form(6.6)dŜ(t)Ŝ(t)=rdt+σρdw*(t).
Using Black-Scholes formula (see [8]) we obtain the European call option pricing formula for our model (6.6):(6.7)C=S0Φ(d+)-Ke-rTΦ(d-),
where (6.8)d+=ln⁡(S0/K)+(r+(1/2)σρt)σρt,d-=ln⁡(S0/K)+(r-(1/2)σρt)σρt,Φ(x) is a normal distribution and σρ is defined in (6.3).


### 6.2. Double Averaged Diffusion GMRP

Using the similar arguments as in (6.1)–(6.7), we can get European call option pricing formula for a double averaged diffusion GMRP in (3.24):(6.9)dŠ(t)Š(t)=12(σ̌ρ2-ρ̌2)dt+σ̌ρdw(t),
where σ̌ρ2 and ρ̌2 are defined in (3.21), (see also (3.13)), and (3.15). Namely, the European call option pricing formula for a double averaged diffusion GMRP is:(6.10)C=S0Φ(d+)-Ke-rTΦ(d-),
where (6.11)d+=ln⁡(S0/K)+(r+(1/2)σ̌ρt)σ̌ρt,d-=ln⁡(S0/K)+(r-(1/2)σ̌ρt)σ̌ρt,Φ(x) is a normal distribution and σ̌ρ is defined in (3.21).


### 6.3. European Call Option Pricing Formula for Merged Diffusion GMRP

From Section3.2, the merged diffusion GMRP has the following form:(6.12)dS̃(t)S̃(t)=12(σ̂ρ2(x̂(t))-ρ̂2(x̂(t)))dt+σ̂ρ(x̂(t))dw(t),
where σ̂ρ2 and ρ̂2 are defined in Section 3.2 (see (3.18). Taking into account the result on European call option pricing formula for regime-switching geometric Brownian motion (see [4, page 224, corollary]), we obtain the option pricing formula for the merged diffusion GMRP:(6.13)C=∫CTBS((zT)2-1,T,S0)FTx(dz),
where CTBS is a Black-Scholes value and FTx(dz) is a distribution of the random variable (6.14)zTx=∫0Tσ̂ρ2(x̂(t))ds,
where x̂(t) is a merged Markov process.


## 6.1. Ergodic Geometric Markov Renewal Process

As we have seen in Section3, an ergodic diffusion GMRP Ŝ(t) satisfies the following SDE (see (3.10)):
(6.1)dŜ(t)Ŝ(t)=12(σρ-ρ̂2)dt+σρdw(t),
where(6.2)ρ̂2=∫Xp(dx)∫XP(x,dy)ρ2(y)m,(6.3)σρ2=∫Xp(dx)[12∫XP(x,dy)ρ2(y)+∫XP(x,dy)ρ(y)R0P(x,dy)ρ(y)m.
The risk-neutral measure P* for the process in (6.1) is:(6.4)dP*P=exp⁡{-θt-12θ2w(t)},
where (6.5)θ=((1/2)(σρ-ρ̂2)-r)σρ.
Under P*, the process e-rtŜt is a martingale and the process w*(t)=w(t)+θt is a Brownian motion. In this way, in the risk-neutral world, the process Ŝt has the following form(6.6)dŜ(t)Ŝ(t)=rdt+σρdw*(t).
Using Black-Scholes formula (see [8]) we obtain the European call option pricing formula for our model (6.6):(6.7)C=S0Φ(d+)-Ke-rTΦ(d-),
where (6.8)d+=ln⁡(S0/K)+(r+(1/2)σρt)σρt,d-=ln⁡(S0/K)+(r-(1/2)σρt)σρt,Φ(x) is a normal distribution and σρ is defined in (6.3).


## 6.2. Double Averaged Diffusion GMRP

Using the similar arguments as in (6.1)–(6.7), we can get European call option pricing formula for a double averaged diffusion GMRP in (3.24):(6.9)dŠ(t)Š(t)=12(σ̌ρ2-ρ̌2)dt+σ̌ρdw(t),
where σ̌ρ2 and ρ̌2 are defined in (3.21), (see also (3.13)), and (3.15). Namely, the European call option pricing formula for a double averaged diffusion GMRP is:(6.10)C=S0Φ(d+)-Ke-rTΦ(d-),
where (6.11)d+=ln⁡(S0/K)+(r+(1/2)σ̌ρt)σ̌ρt,d-=ln⁡(S0/K)+(r-(1/2)σ̌ρt)σ̌ρt,Φ(x) is a normal distribution and σ̌ρ is defined in (3.21).


## 6.3. European Call Option Pricing Formula for Merged Diffusion GMRP

From Section3.2, the merged diffusion GMRP has the following form:(6.12)dS̃(t)S̃(t)=12(σ̂ρ2(x̂(t))-ρ̂2(x̂(t)))dt+σ̂ρ(x̂(t))dw(t),
where σ̂ρ2 and ρ̂2 are defined in Section 3.2 (see (3.18). Taking into account the result on European call option pricing formula for regime-switching geometric Brownian motion (see [4, page 224, corollary]), we obtain the option pricing formula for the merged diffusion GMRP:(6.13)C=∫CTBS((zT)2-1,T,S0)FTx(dz),
where CTBS is a Black-Scholes value and FTx(dz) is a distribution of the random variable (6.14)zTx=∫0Tσ̂ρ2(x̂(t))ds,
where x̂(t) is a merged Markov process.



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*Source: 347105-2010-12-19.xml*