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

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.


Introduction
Let N t be a standard Poisson process and Y k k∈Z be i.i.d. random variables which are independent of N t and S * 0 > 0. The geometric compound Poisson processes 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 2 International Journal of Stochastic Analysis 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 switchedswitching 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 Section 2 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.

The Geometric Markov Renewal Processes (GMRP)
In this section we present the Geometric Markov renewal processes. We closely follow 2 .
Let Ω, B, F t , P be a standard probability space with complete filtration F t and let x k k∈Z be a Markov chain in the phase space X, X with transition probability P x, A , where x ∈ X, A ∈ X. Let θ k 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 : θ k w ≤ x}. The random variables θ k 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 θ k k∈Z we can construct another renewal process τ k k∈Z defined by The random variables τ k are called renewal times or jump times . The process The process x 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 x n , θ n 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 : x v 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 {S t } t∈R as a stochastic functional S t defined by where S 0 > 0 is the initial value of S t . We call this process S t t∈R a geometric Markov renewal process by analogy with the geometric compound Poisson processes where S * 0 > 0, N t is a standard Poisson process, Y k k∈Z are i.i.d. random variables. The geometric compound Poisson processes {S * t } 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 {S t } 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.

Jump Semi-Markov Random Evolutions
Let C 0 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 C 0 R as follows: International Journal of Stochastic Analysis With these contraction operators D x we define the following jump semi-Markov random evolution JSMRE V t of the geometric Markov renewal processes {S t } t∈R in 2.5 : Using 2.7 we obtain from 2.8 where S t is defined in 2.5 and S 0 s. Let Q x, A, t be a semi-Markov kernel for Markov renewal process x n ; θ n n∈Z , that is, where P x, A is the transition probability of the Markov chain x n n∈Z and G x t is defined by 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 value u t, x in 2.10 of the semi-Markov random evolution V t given by the solution of the following Markov renewal equation (MRE)
: where G x t 1 − G x t , G x t : P θ n 1 ≤ t | x n x , g x is a bounded and continuous function on X.

2.13
A detailed information about ρ T x and S T t can be found in Section 4 of 2 . It can be easily shown that 2.14 International Journal of Stochastic Analysis 5 To describe martingale properties of the GMRP S t t∈R in 2.5 we need to find an infinitesimal operator of the process Let γ t : t − τ v t and consider the process x t , γ t on X × R . It is a Markov process with infinitesimal operator The infinitesimal operator for the process ln S t has the form: where the operators A and Q are defined in 2.17 and 2.18 , respectively. Thus we obtain that the process , and the operator A in 2.17 has the form: The process ln S t , x t on R × X is a Markov process with infinitesimal operator It follows that the process

Martingale Property of the GMRP
Consider the geometric Markov renewal processes S t t∈R For t ∈ 0, T let us define where h x is a bounded continuous function such that X h y P x, dy 1, X h y P x, dy p y 0.

2.27
If EL T 1, then geometric Markov renewal process S t in 2.25 is an F t , P * -martingale, where measure P * is defined as follows:

2.28
In the discrete case we have

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 process v t in 2.5 in the new accelerated scale of time tT 2 , that is, v ≡ v tT 2 . Due to more rapid changes of states of the system under the balance condition, the fluctuations are described by a diffusion processes.

Ergodic Diffusion Approximation
Let us suppose that balance condition is fulfilled for functional S T where p x is ergodic distribution of Markov chain x k k∈Z . Then S t S 0 , for all t ∈ R . Consider S T t in the new scale of time tT 2 : Due to more rapid jumps of v tT 2 the process S T t will be fluctuated near the point S 0 as T → ∞. By similar arguments similar to 4.3 -4.5 in 2 , we obtain the following expression: ln Algorithms of ergodic averaging give the limit result for the second term in 3.3 see 1, page 43 and 4, page 88 : where ρ 2 : X p dx X P x, dy ρ 2 y /m. Using algorithms of diffusion approximation with respect to the first term in 3.3 we obtain 4, page 88 : where σ 2 ρ : X p dx 1/2 X P x, dy ρ 2 y X P x, dy ρ y R 0 P x, dy ρ y /m, R 0 is a potential 3, page 68 , of x n n∈Z , w t is a standard Wiener process. The last term in 3.3 8 International Journal of Stochastic Analysis goes to zero as T → ∞. Let S t be the limiting process for S T t in 3.3 as T → ∞. Taking limit on both sides of 3.3 we obtain where σ 2 ρ and ρ 2 are defined in 3.4 and 3.5 , respectively. From 3.6 we obtain S t S 0 e σ ρ w t − 1/2 t ρ 2 S 0 e − 1/2 t ρ 2 e σ ρ w t .

3.7
Thus, S t satisfies the following stochastic differential equation SDE : In this way we have the following corollary.

Corollary 3.1. The ergodic diffusion GMRP has the form
S t S 0 e − 1/2 t ρ 2 e σ ρ w t , 3.9 and it satisfies the following SDE:

Merged Diffusion Approximation
Let us suppose that the balance condition satisfies the following: ρ k X k p k dx X k P x, dy ρ y m k 0, 3.11 for all k 1, 2, . . . , r where x n n∈Z is the supporting embedded Markov chain, p k is the stationary density for the ergodic component X k , 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 : International Journal of Stochastic Analysis 9 where ρ 2 k : X k p k dx X k P x, dy ρ 2 y m k 3.13 using the algorithm of merged diffusion approximation that 1, 3, 4 obtain from the first part of the right hand side in 3.3 : The third term in 3.3 goes to 0 as T → ∞. In this way, from 3.3 we obtain: where S t is the limit S T t as T → ∞. Stochastic differential equation SDE forŠ t has the following form: where x 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 .

Diffusion Approximation under Double Averaging
Let us suppose that the phase space X {1, 2, . . . , r} of the merged Markov process x t consists of one ergodic class with stationary distributions p k ; k {1, 2, . . . r} . Let us also suppose that the balance condition is fulfilled: International Journal of Stochastic Analysis Then using the algorithms of diffusion approximation under double averaging see 3, page 188 , 1, page 49 and 4, page 93 we obtain: and ρ 2 k and σ 2 ρ k are defined in 3.13 and 3.15 , respectively. Thus, we obtain from 3.20 :

Proofs
In this section we present proofs of results in Section 3. 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 S T t in Skorokhod space D R 0, ∞ 7, page 148 ; 2 solution of martingale problem for the limit process S t ; 3 characterization of the limit measure for the limit process S t ; 4 uniqueness of solution of martingale problem. We also give here the rate of convergence in the diffusion approximation scheme.

Diffusion Approximation (DA)
Let where φ 1 f and φ 2 f are defined as follows:

Martingale Problem for the Limiting Problem G 0 t in DA
Let us introduce the family of functions: where φ T are defined in 4.7 and G T j is defined by International Journal of Stochastic Analysis Functions ψ T s, t are F tT 2 -martingale by t. Taking into account the expression 4.6 and 4.7 , we find the following expression:

4.10
where O T −2 is the sum of terms with T −2 nd order. Since ψ T 0, t is F tT 2 -martingale with respect to measure Q T , generated by process G T t in 4.1 , then for every scalar linear continuous functional η s 0 we have from 4.8 -4.10 :

4.11
International Journal of Stochastic Analysis 13 where E T is a mean value by measure Q T . If the process G T tT 2 converges weakly to some process G 0 t as T → ∞, then from 4.11 we obtain that is, the process is a continuous Q T -martingale. Since A is the second order differential operator and coefficient σ 2 1 is positively defined, where 14 then the process G 0 t is a Wiener process with variance σ 2 1 in 4.14 : G 0 t σw t . Taking into account the renewal theorem for v t , namely, T −1 v tT 2 → T → ∞ t/m, and the following representation we obtain, replacing tT 2 by v tT 2 , that process G T t converges weakly to the process G 0 t as T → ∞, which is the solution of such martingale problem: is a continuous Q T -martingale, where A 0 : A/m, and A is defined in 4.5 -4.5 .

Weak Convergence of the Processes G T t in DA
From the representation of the process G T t it follows that

International Journal of Stochastic Analysis
This representation gives the following estimation: Taking into account the same reasonings as in 2 we obtain the weak convergence of the processes G T t in DA. we obtain that there exists the measure Q on D R 0, ∞ which solves the martingale problem for the operator A 0 or, equivalently, for the process G 0 t in the form 4.12 . Uniqueness of the solution of the martingale problem follows from the fact that operator A 0 generates the unique semigroup with respects to the Wiener process with variance σ 2 1 in 4.14 . As long as the semigroup is unique then the limit process G 0 t is unique. See 3, Chapter 1 .