Feller Property for a Special Hybrid Jump-Diffusion Model

Abstract

We consider the stochastic stability for a hybrid jump-diffusion model, where the switching here is a phase semi-Markovian process. We first transform the process into a corresponding jump-diffusion with Markovian switching by the supplementary variable technique. Then we prove the Feller and strong Feller properties of the model under some assumptions.

1. Introduction

Stability of stochastic differential equations with Markovian switching has received a lot of attention. Assume that is a two-component Markov process such that is a continuous component taking values in and is a Markov process taking values in a finite set. More specially, the process can be described by

Mao [1] investigated the exponential stability for general nonlinear stochastic differential equations with Markovian switching. Shaikhet [2] took the time delay into account and considered the stability of a semilinear stochastic differential delay equation with Markovian switching, while Mao et al. [3] investigated the stability of a nonlinear stochastic differential delay equation with Markovian switching.

Most of these papers are concerned with asymptotic stability in probability or in mean square (i.e., the solution will tend to zero in probability or in mean square). However, this asymptotic stability is sometimes too strong and in this case it is useful to know whether or not the solution will converge in distribution (not necessary to converge to zero). Hence Wee [4] studied the stability in distribution for jump-diffusions (without Markovian switching), whereas Yuan and Mao [5] made many corresponding researches for continuous Itô diffusions with Markovian switching. Zhang and Chen [6] considered a general condition for existence and uniqueness of stationary distribution for diffusion systems.

In addition, Xi [7] recently discussed the Feller continuity of a general nonlinear stochastic differential equation with Markovian switching. Later, Xi [8] further considered the Feller continuity and exponential ergodicity for a jump-diffusion equation with Markovian switching of the form where is -matrix valued and and are valued for and . is a -dimensional standard Brownian motion and is a Poisson measure.

Note that these papers referred to above are all concerned with Markovian switching systems. However, we know that Markovian switching systems have many limitations in applications. For example, the jump time of a Markov chain is, in general, exponentially distributed. And it is well known that the pervasiveness of the exponential distribution in stochastic systems is rarely due to empirical evidence in support of their assumption, but far more so to the ease of conditioning which results from the lack of memory property. Hence the results obtained on these systems are conservative. Besides, it is worthy to recall the phase type distribution which is a generalization of exponential distribution while still preserving much of its analytic tractability. The phase type distribution is very important in real application since the matrix-analysis method developed by Neuts [9].

Motivated by the analysis above, our aim in this paper is to establish some criteria on the stochastic stability for a kind of more general jump-diffusions with phase semi-Markovian switching. The rest of the paper is organized as follows. In Section 2, we begin with the review of certain notions for phase type distribution and phase semi-Markov process. And then, through the supplementary variable technique, we make a transformation for the process considered in this paper. In Section 3, by using the coupling method, a proof of Feller continuity is given. Furthermore, we also prove the strong Feller continuity in Section 4.

2. Preliminaries and Transformation

Throughout the paper, let be a probability space supporting all the random variables and processes and an increasing family of sub--algebras of . To proceed, we will recall some definitions and properties.

Definition 1. A probability distribution on is said to be a continuous phase type distribution, if it is the distribution of the lifetime of a terminating Markov process with finitely many states and time homogeneous transition rates.

More precisely, a terminating Markov process with state-space and intensity matrix is defined as the restriction to of a Markov process on , where is absorbing and the states in are transient (we often write for the number of elements of ). For any initial distribution , is the distribution of the time to absorption, . This implies in particular that the intensity matrix for can be written in block-partitioned form as where the matrix satisfies , , ; is a nonnegative column vector.

The pair is called the order representation of . A basic analytical property of phase-type distribution is given by the following result.

Lemma 2. Let be a phase type with representation ; then, for , the cumulative distribution function is given by

Definition 3. Let be a finite or countable set. A stochastic process on the state-space is called a phase semi-Markov process (when is finite, is also called a finite phase semi-Markov process), if the followings hold.(1)The sample paths of are right-continuous step functions and have left-handed limits with probability one.(2)Denote the th jump point of the process by , where , , and the process possesses Markov property at each .(3) does not depend on and .(4) is a phase type distribution.

In the present paper, the process considered can be described by where is -matrix valued and and are valued for and . Let be a finite phase semi-Markov process on the state-space , and let be an -adapted -valued Brownian motion and also a martingale with respect to ; Let be a stationary -Poisson point process, and let be the compensated Poisson random measure on , where is a deterministic finite characteristic measure on the measurable space . Assume that the Brownian motion , the Poisson process , and the semi-Markov process are independent from each other.

From the description above, it is easy to see that the process is not a Markov process, unless the sojourn time of in each state is exponentially distributed. But we can show that considering the process only at the jump points yields a (discrete-time) Markov process. Furthermore, the behaviour of is piecewise deterministic in the intervals between jump points. For the Markovization of we therefore have to add the information on the neighboring jump points. Hence, to continue with our study, let denote the phase of at time ; then is a Markov process, moreover, so is process .

Next, we will show some properties of Markov process . Denote the th jump point of the finite phase semi-Markov process by , where . From Definition 3, we know that the staying time at each state is of phase type. Hence, for each , correspondingly, we let denote the order representation of and the transient states set, where denotes the number of the elements in , and Let For any , define We will see that the probability distribution of can be determined only by .

Remark 4. The superscript of parameters , and so forth and the subscript of all mean that these parameters are corresponding to the state .

Theorem 5. Process is a Markov process with state-space ( is finite if and only if E is finite). Then the infinitesimal generator of given by is determined only by the pair of given by as follows:

Proof. For any , one has the following:(1)for any , we have (2)for any , , and , we have (3)for any , , and , we have The proof is complete.

For the existence and uniqueness of process satisfying (5), we make the following assumption.

Assumption 6. Assume that is measurable function and that and are continuously differentiable in , and they satisfy the Lipschitz condition and the linear growth condition as follows. For some constant , (A1)′(A2)′for all and .

Theorem 7. Under Assumption 6, system (5) has a unique solution .

Proof. From Theorem 9.1 in Chapter IV of Ikeda and Watanabe [10], it follows that, for each , there exists a unique strong solution to the following stochastic differential-integral equation: such that is a -measurable random variable and is a family of -adapted jump-diffusions. Since and are mutually independent, they have no common jumps almost surely. Therefore, each will not coincide with any jump instant of for all and we can construct the process as follows. For any given initial condition , on the interval , set Next, as was done in Section 9 in Chapter IV of Ikeda and Watanabe [10], set , , and , where is the random point function corresponding to . Similarly, we can determine the process on the time interval with respect to as above. Then, define Continuing this procedure successively, is determined uniquely on the interval for every and thus is determined globally due to .
Process is the associated Markov process of . And for any , and , we define functions , , and as follows:
It is easy to show that for any Consequently, we have the following result.

Theorem 8. The process defined in (5) is equivalent to the following process: where functions , and are defined in (17)–(19). is the associated Markov process of phase semi-Markov process with infinitesimal generator defined in (9), and processes , , and are defined in process (5).

Proof. Substitute (17)–(19) into (20); then the theorem can be proved easily following the identity relation between functions , and and , and , respectively.
Based on Assumption 6 and (17)–(19), we can easily get the following proposition.

Proposition 9. If Assumption 6 holds for functions , and , then we have that is measurable function and that and are continuously differentiable in , and they satisfy the Lipschitz condition and the linear growth condition as follows: For some constant ,(A1) , (A2)for all and .

In the following two sections, we turn to discuss the stochastic stability of the Markov process with state-space satisfying (9) and (20).

3. Feller Property

Let and denote the inner product and the gradient operator in , respectively. If is a vector or matrix, we use to denote its transpose. For , set . Define a metric on as follows: where , and Therefore, is a complete separable metric space and we then have a natural càdlàg space .

Process has a generator as follows. For each and for any twice continuously differentiable functions , Here operators , , and are further defined as follows:

Lemma 10. Suppose that Assumption 6 holds. Then the jump-diffusion process is nonexplosive. (For the proof please see Xi [8].)

In the following, we prove the Feller continuity of process by using the coupling method. First, we construct a coupling for the generator as follows: for functions on , which are twice continuously differentiable in both and and have compact support. The couplings , , and are, respectively, the corresponding parts of , , and in operator , which can be constructed as follows: where is a bounded function on . Let be the Markov chain generated by the coupling operator . Set ; then will move together from onward. Next, for and , set Obviously, is nonnegative definite for all and . According to Chen and Li [11], the diffusion operator is determined by and , which is a coupling of and . Using the change of variable theorem, we can rewrite the operator as follows: where for , ; and . Here actually is the jump measure of the first component of . On the set of twice continuously differentiable functions of to with compact support, set then the operator is a coupling of and .

Next, we denote the transition probability family of the process by . For a subsequent use, we now introduce the Wasserstein metric between two probability measures as follows. For two probability measures and , define where varies over all coupling probability measures with martingales and . In order to prove the Feller continuity of , we need to make the following assumption.

Assumption 11. Assume that there exists such that for all and , where denotes the total variation norm.

Theorem 12. Let be the solution to the system given by (5) with initial value . Assume that Assumptions 6 and 11 hold. Then for any bounded continuous function , the function is continuous with respect to .

Proof. Recall that the is a Markov process associated with the phase semi-Markov process . Hence , where . To prove the Feller property for , it is equivalent to show Feller property for with respect to . Since has the discrete metric, we only need to prove that, for any , , and , converges weakly to as . To this end, by means of Theorem 5.6 in Chen [12], it suffices to prove that We now use the coupling constructed in (25) to prove (33). Let denote the process corresponding to the coupling generator , let denote the distribution of starting from , and let denote the corresponding expectation. Set Using Dynkin’s formula and noting that for all whenever , we have for and .
Next, we give some estimation on the integrand of the rightmost term in (35). It follows from (18) in Chen and Li [11] that where Moreover, Therefore, from (36) and Proposition 9, one has for and . With some straightforward calculations, we have for and .
Now substituting (39) and (40) into (35), we arrive at Thus, by Gronwall’s inequality, we obtain Finally, letting and , we conclude that which implies (33). The proof is complete.

4. Strong Feller Continuity

In the following we will prove the strong Feller continuity of the process by using the relation between the transition probability of jump-diffusion and the corresponding diffusions in Skorohod [13]. First we introduce some auxiliary processes as follows. For each , let be the unique strong solution to the following stochastic differential equation in : For each , we denote the transition probability families of the jump-diffusion satisfying (20) when and the diffusion by and , respectively.

Lemma 13. For any given , if the transition probability has a density with respect to the Lebesgue measure, then the transition probability also has a density with respect to Lebesgue measure. Moreover, if the transition probability density is positive, then the transition probability density also is positive.

Proof. Please see Chapter 1 in Skorohod [13].

Assumption 14. For each , assume that the diffusion determined by (44) has a positive probability density with respect to the Lebesgue measure.

To prove the strong Feller continuity, we need to prove that, for every and every , the transition probability corresponding to is absolutely continuous with respect to a reference measure . Here and hereafter, the reference measure is the product measure on of the Lebesgue measure on and the counting measure on . In addition, the Lebesgue measure on will be denoted by . Hereafter, for a given set , let, for each , be the section of at . Then, the product measure theorem gives us the following lemma which will be used later.

Lemma 15. For any set with , we have that for all .

Recall that . Therefore, for every , every , and every set , we have that

Lemma 16. Suppose that Assumption 14 holds. One has the following.(i)For every , every , and every integer , we have that  for any set satisfying .(ii)For every and every , we have that  for any set satisfying .

The following theorem is our main result in this section.

Theorem 17. Suppose that Assumptions 11 and 14 hold. For every and every , the transition probability is absolutely continuous with respect to . Then, for any bounded measurable function , the function is continuous with respect to .

Proof. The former assertion readily follows from Lemma 16. In the following we prove the latter assertion. For a set , let , , and denote the closure, the interior, and the boundary of , respectively. Following from the definition of the reference measure, it is easy to get that for all . Therefore, by virtue of the absolute continuity and the Feller continuity proved in Theorem 12 and Lemma 16, we can conclude that, for every and every , which implies the desired strong Feller continuity. The proof is therefore complete.

5. Conclusion

In this paper, we discuss the stochastic stability of a more general jump-diffusion process with phase semi-Markovian switching. We first transform it into a Markov process by the way of supplementary variable technique. Then using coupling method, we verify that, under linear growth and some proper conditions, the jump-diffusion with phase semi-Markovian switching keeps Feller continuous and strong Feller continuous. However, it is worthy to study some other properties (recurrence property, existence and uniqueness for the stationary distribution) for the process defined by (5).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the anonymous referees for their valuable comments and suggestions which led to improvements in this paper. Research of the author was partially supported by National Natural Science Foundation of China (nos. 11171062, 11101077 and 11201062), the Fundamental Research Funds for the Central Universities.