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Numerical Solution of Nonlinear Stochastic Differential Delay Equation with Markovian Switching

Chaozhu Hu*, Shaobo Zhou

School of Science, Hubei University of Technology, Wuhan 430068, Hubei, China

Abstract: This paper is concerned with the Euler-Maruyama approximate solution of nonlinear stochastic delay differential equations with Markovian switching (SDDEwMSs). We establish the existence and uniqueness results for the global solution of SDDEwMSs under the polynomial growth and the local Lipschitz condition. we then introduce Euler-Maruyama approximate solution of this equation, and establish the convergence in probability of the numerical solution to the exact solution of the problem without the linear growth condition. As an application, we also give one example to demonstrate our results.

Keywords: stochastic differential delay equation, Euler-Maruyama, convergence in probability, markovian switching

Citation: Chaozhu Hu, Shaobo Zhou, Numerical Solution of Nonlinear Stochastic Differential Delay Equation with Markovian Switching. Journal of Mathematical Sciences and Applications, 4 (1). 20-28. January 2016. doi:10.12691/jmsa-4-1-4

1. Introduction

Stochastic modelling has come to play an important part in many areas of science and engineering for a long time. Most of stochastic modelling cannot be solved explicitly. As a result, numerical and analytical techniques have been used to study such problems. Many numerical techniques designed to produce approximate solutions in the literature, see for example [2-7] and references therein. It is well known that the local Lipschitz and the linear growth condition are classical conditions in order to guarantee existence and uniqueness of the global solutions (see [1]). In [2,3,4], the authors investigate the convergence and stability of Euler-Maruyama numerical solution under these classical conditions. However, many stochastic equations do not satisfy the linear growth condition. Recently, In [5], the author consider an even more general Khasminskii-type test for nonlinear stochastic delay differential equations (SDDEs) that covers a wide class of highly nonlinear equations, and studied convergence in probability of the Euler-Maruyama solution for SDDEs. For the similar result of highly nonlinear neutral stochastic delay differential equations (NSDDEs) with time-dependent delay, please see Milosevic [6]. Specially, that Zhou and Fang [7] established new criteria of the existence-and-uniqueness of the global solution and the convergence in probability of Euler-Maruyama approximate solution for nonlinear NSFDEs under the polynomial growth conditions.

On the other hand, we also remark that a great deal of reseach for the stochastic differential equations (SDEs) are successfully extended to the stochastic differential equations with Markovian switching (SDEwMSs)and the stochastic delay differential equations with Markovian switching (SDDEwMSs) (see [8,9,10,11]). In [9], the authors introduce the Euler-Maruyama(EM) numerical solution which strong converge to the actual solution under the global Lipschitz condition, and the same problem have been discussed under the local Lipschitz condition and the linear growth condition, and furthermore describe the convergence in probability, instead of L², under some additional conditions in terms of Lyapunov-type functions. In [12], the numerical solution for NSDDEwMSs are discussed. And in [13], the authors consider the strong convergence in the sense of the Lp-norm when the drift and diffusion coefficients are Taylor approximations. However, to the best of our knowledge, few papers can be found in the literature on the numerical methods for nonlinear SDDEwMSs. So, being directly inspired by [7], the purpose of this paper to study the Numerical solution of nonlinear stochastic differential delay equation with Markovian switching.

The paper is organized as follows: Some necessary notations and the property of right-continuous Markov chain are in Section 2. In Section 3, we prove the existence and uniqueness of the solution of SDDEwMSs under the polynomial growth. In Section 4, the Euler-Maruyama approximate solution for SDDEwMSs are obtained, and establish the convergence in probability. Finally, in Section 5, we give one example to demonstrate our results.

2. Preliminaries

Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous and contains all P-null sets). Let be an m-dimensional Brownian motion, and let be a right-continuous Markov chain on the probability space taking values in a finite state space Set be the Euclidean norm in If is a vector or matrix, its transpose is denoted by If is a matrix, its trace norm is denoted by while its operator norm is denoted by For we shall denote by the family of continuous functions from to with the norm Let denote by the family of all -measurable and -valued random variables such that

Set be a right-continuous Markov chain on the probability space taking values in a finite state space with the generator given by

where > 0. Here > 0 is transition rate from i to j if , while

We assume that the Markov chain is independent of the Brownian motion w(t). It is well know that almost every sample path of is a right-continuous step function with finite number of simple jumps in any finite subinterval of R₊ = [0,∞). Moreover, for convenience, denoted by y(t) = x(t − τ).

Consider the d-dimensional Euler-Maruyama (EM) numerical solutions of nonlinear stochastic differential delay equation with Markovian switching (SDDEwMSs)

[picture] | (2.1)

with the initial data Here

Assumption (Local Lipschitz Condition) For each integer and there exists a positive constant , such that

[picture] | (2.2)

for with

Remark 2.1. According to assumption (H₁), it is easy to obtain that

[picture] | (2.3)

Similarly, |g(x, y, i, t)|² ≤ KR(1 + |x|² + |y|²). Here,

Assumption (H₂). (Polynomial Growth Condition) Assume that for some positive integer L, there exist positive constants a₁, a₂, a₃, b₁, b₂, b₃, α, β, such that

[picture] | (2.4)

Let denote the family of all nonnegative functions V (x, i, t) on which are continuously twice differentiable in x and once differentiable in t. If define an operator LV from to R by

[picture] | (2.5)

where

In particular, if V is independent of i, that is V (x, i, t) = V (x, t), then

[picture] | (2.6)

since

Let by the generalized Itȏ formula, we obtain:

holds for any stopping times 0 ≤ ≤ < ∞ as long as the integrations involved exist and are finite.

3. Global Solution of SDDEwMSs

In this sections, we prove the existence and uniqueness of the solution of SDDEwMSs under the polynomial growth.

Theorem 3.1. Let Assumptions (H₁) and (H₂) hold, and assume that then, for any initial condition there almost surely exists a unique global solution x(t, ξ) to equation (2.1) on t ≥ −τ. Moreover, there exists a positive constant such that

Proof. Bearing in mind the local Lipschitz condition (2.2), it follows that for any given initial data there exists a unique maximal local solution to Eqs. (2.1), where is the explosion time. To show this solution is global, we only need to show that a.s.

Assume that there exists an integer such that For each integer define the stopping time

[picture] | (3.1)

and (as usual, ϕ =the empty set). Define it is obvious that is an increasing function with k, so a.s. Our goal is to prove that a.s, which implies that In other words, we only prove that

Define V (x, i, t) = |x|p, we obtain

[picture] | (3.2)

Using the Assumption and the inequality we can obtain

[picture] | (3.3)

where

Recalled that conditions of this theorem, there exists a positive constant c₀, such that

[picture] | (3.4)

Substituting for this an (3.4) into (3.3), implies

Therefore

The Gronwall inequality implies

Similar, we have By the definition of , we have that

Clearly, we have that P(τk ≤ t) → 0(k → ∞, t > 0).

4. Euler-Maruyama Method

In this section, we define the Euler-Maruyama approximate solution.

Lemma 4.1. ([12]) Given for and , then is a discrete Markov chain with the one-step transition probability matrix

Since the γij are independent of x, the paths of r can be generated independently of x and in fact, before computing x.

Let a stepsize with satisfies for some positive integer M. Define for the discrete markovian chain can be simulated as follows: Compute the one-step transition probability matrix Let and generate a random number ζ1 which is uniformly distributed in [0, 1]. Define

where we set as usual. Generate independently a new random number which is again uniformly distributed in [0, 1] and then define

Repeating this procedure, a trajectory of can be generated. This procedure can be carried out independently to obtain more trajectory.

After explaining how to simulate the discrete Markov chain the Euler-Maruyama numerical scheme applied to the Eqs.(2.1) is to compute the discrete approximations by setting for and forming

[picture] | (4.1)

where

For each define with the initial value on That is

[picture] | (4.2)

while the continuous-time Euler-Maruyama approximation process X(t) on is to be interpreted as the stochastic integral.

[picture] | (4.3)

Therefore

[picture] | (4.4)

It is useful to know that that is, and concide with the discrete approximate solution at the gridpoints. For convenience, let T > 0 be arbitrary and define the sequence of stopping times

and

For convenience, let C be a positive constant independent of h, and the product of C and other constants is still denoted by C.

Lemma 4.2 Under Assumptions (H₁),

Proof. Recalling the Lemma 3.1 in [7]. By (4.3) we have

[picture] | (4.5)

Using the Holder inequality and (2.3)

[picture] | (4.6)

By the BDG inequality

[picture] | (4.7)

Using (4.7)and (4.6), the estimate becomes

The Gronwall inequality gives

Lemma 4.3 Let Assumption (H₁) hold, for any , there exist a positive constant C independent of h, such that

Proof. We have by definition of and , thus

Then by Lemma 4.2, it is easy to obtain that thus

Lemma 4.4 Under Assumption and for any and there exists a sufficiently large and sufficiently small such that

Proof. Recalling that (3.2), the proof of this lemma is similar to the argument of Theorem 3.5 in [7]. Applying the generalized Itȏ formula to yields.

where

Let R be sufficiently large integer, if then, the Assumption (H₁) implies there exists constant C which depends on R such that

Hence

[picture] | (4.8)

By the Lemma 4.2 and we have that

[picture] | (4.9)

Recalling the proof of Theorem 3.1,

here be a positive constant. Repeating the procedure from Theorem 3.1, we can prove that which completes the proof.

Lemma 4.5 [12] Let the Assumption (H₁) hold, for every we have

[picture] | (4.10)

[picture] | (4.11)

C is a positive constant dependent on max₀≤i≤N(−γii), but independent of h.

Lemma 4.6 Under the condition of Theorem 3.1, the numerical solution convergence to the exact solution of Eqs.(2.1) in the sense

Proof. From Eqs.(2.1) and Eqs.(4.3), we have

By the inequality (a + b)² ≤ 2a² + 2b², for any ,

[picture] | (4.12)

By the Holder inequality, Lemma 4.3, Assumption (H₁), and (4.10), we have

[picture] | (4.13)

Similarly, by the Burkholder-Davis-Gundy inequality , Lemma 4.3 and (4.11), we may obtain

[picture] | (4.14)

Substituting (4.13) − (4.14) into (4.12), yields

The Gronwall inequality implies

that is,

The proof is completed.

Theorem 4.7. Under Assumption (H1), (H2), for arbitrary T > 0.

Proof. For arbitrary ϵ ∈ (0, 1), we define

If we can show that P(B) ≤ ϵ, then the Euler-Maruyama approximate solution converges to the exact solution of Eqs. (2.1).

Recalling the proof of Theorem 3.1 and Lemma 3.4, there exists a sufficient large R* = R(ϵ, T) and sufficiently small such that

By Lemma 4.6, for the sufficiently small h, we have

Therefore

The proof is completed.

5. One Example

In this section, in order to illustrate our results, we consider an numerical example.

Example. Consider the following scalar nonlinear stochastic differential equation with Markovian switching

[picture] | (5.1)

on t ≥ 0, where, w(t) is a scalar Brownian motion, r(t) is a right continuous Markov chain taking values in with the generator

of course w(t) and r(t) are assumed to be independent, and a₁, a₂, a₃, b₁ are positive constants. It is to obtain

By the Theorem 3.1, we assume that

and

then Eqs. (5.1) has unique global solution.

To carry out the numerical simulation we choose the step size h = 1/1024, and a(1) = 0.1, a(2) =0.05, b(1) = 0.05, b(2) = 0.02, the computer simulation result is shown in Figure 1 and Figure 2. And it is clear that the Euler-Maruyama method reveals the almost surely exponentially stable property of the solution.

Figure 1. a₁ = 0.8, a₂ = 0.5, a₃ = 1.2, b₁ = 0.2

Figure 2. a₁ = 0.6, a₂ = 1, a₃ = 1.8, b₁ = 0.1

References

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