Abstract and Applied Analysis

Volume 2013 (2013), Article ID 752953, 12 pages

http://dx.doi.org/10.1155/2013/752953

Research Article

Stationary in Distributions of Numerical Solutions for Stochastic Partial Differential Equations with Markovian Switching

Yi Shen¹ and Yan Li1,2

¹Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

²College of Science, Huazhong Agriculture University, Wuhan 430079, China

Received 30 December 2012; Accepted 24 February 2013

Academic Editor: Qi Luo

Copyright © 2013 Yi Shen and Yan Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate a class of stochastic partial differential equations with Markovian switching. By using the Euler-Maruyama scheme both in time and in space of mild solutions, we derive sufficient conditions for the existence and uniqueness of the stationary distributions of numerical solutions. Finally, one example is given to illustrate the theory.

1. Introduction

The theory of numerical solutions of stochastic partial differential equations (SPDEs) has been well developed by many authors [1–5]. In [2], Debussche considered the error of the Euler scheme for the nonlinear stochastic partial differential equations by using Malliavin calculus. Gyöngy and Millet [3] discussed the convergence rate of space time approximations for stochastic evolution equations. Shardlow [5] investigated the numerical methods of the mild solutions for stochastic parabolic PDEs derived by space-time white noise by applying finite difference approach.

On the other hand, the parameters of SPDEs may experience abrupt changes caused by phenomena such as component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbances [6–9], and the continuous-time Markov chains have been used to model these parameter jumps. An important equation is a class of SPDEs with Markovian switching

Here the state vector has two components and , the first one is normally referred to as the state while the second one is regarded as the mode. In its operation, the system will switch from one mode to another one in a random way, and the switching among the modes is governed by the Markov chain .

Since only a few SPDEs with Markovian switching have explicit formulae, numerical (approximate) schemes of SPDEs with Markovian switching are becoming more and more popular. In this paper, we will study the stationary distribution of numerical solutions of SPDEs with Markovian switching. Bao et al. [10] investigated the stability in distribution of mild solutions to SPDEs. Bao and Yuan [11] discussed the numerical approximation of stationary distribution for SPDEs. For the stationary distribution of numerical solutions of stochastic differential equations in finite-dimensional space, Mao et al. [12] utilized the Euler-Maruyama scheme with variable step size to obtain the stationary distribution and they also proved that the probability measures induced by the numerical solutions converge weakly to the stationary distribution of the true solution. But since the mild solutions of SPDEs with Markovian switching do not have stochastic differential, a significant consequence of this fact is that the Itô formula cannot be used for mild solutions of SPDEs with Markovian switching directly. Consequently, we generalize the stationary distribution of numerical solutions of the finite dimensional stochastic differential equations with Markovian switching to that of infinite dimensional cases.

Motived by [11–13], we will show in this paper that the mild solutions of SPDE with Markovian switching (1) have a unique stationary distribution for sufficiently small step size. So this paper is organised as follows: in Section 2, we give necessary notations and define Euler-Maruyama scheme of mild solutions. In Section 3, we give some lemmas and the main result in this paper. Finally, we will give an example to illustrate the theory in Section 4.

2. Statements of Problem

Throughout this paper, unless otherwise specified, we let be complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). Let be a real separable Hilbert space and an -valued cylindrical Brownian motion (Wiener process) defined on the probability space. Let be the indicator function of a set . Denote by and the family of bounded linear operators and Hilbert-Schmidt operator from into , respectively. Let , , be a right-continuous Markov chain on the probability space taking values in a finite state space with the generator given by

where . Here is the transition rate from to if while

We assume that the Markov chain is independent of the Brownian motion . It is well known that almost every sample path of is a right-continuous step function with finite number of simple jumps in any finite subinterval of .

Consider SPDEs with Markovian switching on

with initial value and . Here , . Throughout the paper, we impose the following assumptions.

(A1) is a self-adjoint operator on generating a -semigroup , such that for some . In this case, has discrete spectrum with corresponding eigenbasis of . (A2) Both and are globally Lipschitz continuous. That is, there exists a constant such that

(A3) There exist and ,   such that

It is well known (see [1, 8]) that under (A1)–(A3), (4) has a unique mild solution on . That is, for any and , there exists a unique -valued adapted process such that

Moreover, the pair is a time-homogeneous Markov process.

Remark 1. We observe that (A2) implies the following linear growth conditions:

where .

Remark 2. We also establish another property from (A3):

where and for .

Denote by the mild solution of (4) starting from . For any subset , let be the probability measure induced by . Namely,

where is the family of the Borel subset of .

Denote by the family by all probability measures on . For , define the metric as follows:

where , and , for .

Remark 3. It is known that the weak convergence of probability measures is a metric concept with respect to classes of test function. In other words, a sequence of probability measures of converges weakly to a probability measure if and only if .

Definition 4. The mild solution of (4) is said to have a stationary distribution if the probability measure converges weakly to as for every , and every , a bounded subset of , that is,

By Theorem 3.1 in [10] and Theorem 3.1 in [14], we have the following.

Theorem 5. Under (A1)–(A3), the Markov process has a unique stationary distribution .

For any , let be the orthogonal projection. Consider SPDEs with Markovian switching on ,

with initial data ,  . Here ,  ,  .

Therefore, we can observe that

By the property of the projection operator and (A2), we have

Hence, (13) admits a unique strong solution on (see [8]).

We now introduce an Euler-Maruyama based computational method. The method makes use of the following lemma (see [15]).

Lemma 6. Given , then is a discrete Markov chain with the one-step transition probability matrix

Given a fixed step size and the one-step transition probability matrix in (16), the discrete Markov chain can be simulated as follows: let , and compute a pseudorandom number from the uniform distribution.

Define

where we set as usual. Having computed , we can compute by drawing a uniform pseudorandom number and setting

The procedure can be carried out repeatedly to obtain more trajectories.

We now define the Euler-Maruyama approximation for (13). For a stepsize , the discrete approximation , is formed by simulating from , and

where .

To carry out our analysis conveniently, we give the continuous Euler-Maruyama approximation solution which is defined by

where and denotes the integer part of and , and .

It is obvious that coincides with the discrete approximation solution at the gridpoints. For any Borel set , let ,

Following the argument of Theorem 5 in [13], we have the following.

Lemma 7. is a homogeneous Markov process with the transition probability kernel .

To highlight the initial value, we will use notation .

Definition 8. For a given stepsize , is said to have a stationary distribution if the -step transition probability kernel converges weakly to as , for every , that is,

We will establish our result of this paper in Section 3.

Theorem 9. Under (A1)–(A3), for a given stepsize , and arbitrary ,  , has a unique stationary distribution .

3. Stationary in Distribution of Numerical Solutions

In this section, we shall present some useful lemmas and prove Theorem 9. In what follows, is a generic constant whose values may change from line to line.

For any initial value , let be the continuous Euler-Maruyama solution of (20) and starting from . Let be the mild solution of (4) and starting from .

Lemma 10. Under (A1)–(A3), then

Proof. Write ,  . From (20), we have

Thus,

Then, by the Hölder inequality and the Itô isometry, we obtain

From (A1), we have

here we use the fundamental inequality ,  . And, by (8), it follows that

Substituting (27) and (28) into (26), the desired assertion (23) follows.

Lemma 11. Under (A1)–(A3), if , then there is a constant that depends on the initial value but is independent of , such that the continuous Euler-Maruyama solution of (20) has

where ,  .

Proof. Write ,  . From (20), we have the following differential form:

with .

Let . By the generalised Itô formula, for any , we derive from (30) that

By the fundamental transformation, we obtain that

By Höld inequality, we have

Then, from (31), we have

By the elemental inequality: ,  ,  , and (8), (27), we obtain that, for ,

By (A2) and (8), we have

Similarly, we have

Thus, we obtain from (36) that

By Markov property, we compute

where . Substituting (39) into(38) gives

Furthermore, due to (37) and (39), we have

On the other hand, by Lemma 10, when , we have

Putting (35), (40), and (41) into (34), we have

By Lemma 10 and the inequality (42), we obtain that

Let , for , then

That is,

Lemma 12. Let (A1)–(A3) hold. If , then

where is a bounded subset of .

Proof. Write ,  ,  . From (20), it is easy to show that

By using the argument of Lemma 10, we derive that, if ,

If , then

Using (30) and the generalised Itô formula, for any , we have

By the fundamental transformation, we obtain that

By the Höld inequality, we have

Then, from (52) and (A3), we have

By (A2) and (27), we have, for ,

It is easy to show that

By (39), we have

Therefore, we obtain that

On the other hand, using the similar argument of (58), we have

Hence, we have

By (50), we obtain that

Let , for , then the desired assertion (47) follows.

We can now easily prove our main result.

Proof of Theorem 9. Since is finite-dimensional, by Lemma 3.1 in [12], we have

uniformly in .

By Lemma 7, there exists , such that

By the triangle inequality (63) and (64), we have

4. Corollary and Example

In this section, we give a criterion based -matrices which can be verified easily in applications.

(A4) For each , there exists a pair of constants and such that, for ,

 Moreover, is a nonsingular -matrix [8].

Corollary 13. Under (A1), (A2), and (A4), for a given stepsize , and arbitrary , has a unique stationary distribution .

Proof. In fact, we only need to prove that (A3) holds. By (A4), there exists , such that .

Set , by (66), we have

In the following, we give an example to illustrate the Corollary 13.

Example 14. Consider

We take and with domain , then is a self-adjoint negative operator. For the eigenbasis , ,  ,  . It is easy to show that

This further gives that

where , thus (A1) holds.

Let be a scalar Brownian motion, let be a continuous-time Markov chain values in , with the generator

Then , .

Moreover, satisfies

where ,  .

Defining , then

It is easy to compute

So the matrix becomes

It is easy to see that is a nonsingular -matrix. Thus, (A4) holds. By Corollary 13, we can conclude that (68) has a unique stationary distribution .

Acknowledgment

This work is partially supported by the National Natural Science Foundation of China under Grants 61134012 and 11271146.

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