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Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching



Hua Yang and Feng Jiang
 
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ABSTRACT

Stability of stochastic systems with Markovian switching has come to play an important role in information science and engineering. The aim of the study is to discuss the stability of the semi-implicit Milstein scheme of stochastic differential delay equations with Markovian switching. The conditions of the General Mean-square (GMS) stability and Mean-square (MS) stability of the semi-implicit Milstein scheme are given by means of the conditions of the analytical solution. The obtained result shows that the numerical scheme reproduces the stability of the analytical solution to stochastic differential delay equations with Markovian switching under some conditions.

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  How to cite this article:

Hua Yang and Feng Jiang, 2014. Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching. Information Technology Journal, 13: 1463-1466.

DOI: 10.3923/itj.2014.1463.1466

URL: https://scialert.net/abstract/?doi=itj.2014.1463.1466
 
Received: May 31, 2013; Accepted: June 07, 2013; Published: March 20, 2014



INTRODUCTION

Hybrid systems have come to play an important role in information science, engineering and mechanics (Mariton, 1990; Huang et al., 2007; Lou and Cui, 2009; Zhu et al., 2010). One of the important classes of the hybrid systems is the stochastic differential delay equations with Markovian switching (SDDEsMS):

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching
(1)

where, r(t), t≥0 be a right-continuous Markov chain on the probability space.

In general, explicit solutions can hardly be obtained for system (1). Thus, it is necessary to develop appropriate numerical methods and to study the properties of these approximate schemes. Stability of numerical Schemes for Stochastic Differential Delay Equations (SDDEs) is essential to avoid a possible explosion of numerical solutions. The convergence and stability properties of the numerical methods for the stochastic ordinary differential equations have been studied by many authors (Mao, 2007; Higham et al., 2002; Hu and Huang, 2011; Zhou and Wu, 2009; Cao et al., 2004; Wang and Zhang, 2006). Mao and Yuan discussed systematically the existence and stability of solutions for stochastic differential equations with Markovian switching (Mao and Yuan, 2006). Rathinasamy and Balachandran (2008) studied the convergence and stability of the semi-implicit Euler-Maruyama method to linear SDDEsMS. Jiang et al. (2011) gave the conditions of stability of analytical solutions and the split-step backward Euler method to linear delay stochastic integro-differential equations with Markovian switching. In this study, the linear stochastic differential delay equations with Markovian switching is studied. The main aim of the study is to extend to SDDEsMS and study the General Mean-square (GMS) stability and Mean-square (MS) stability of the semi-implicit Milstein numerical approximations.

STABILITY OF ANALYTICAL SOLUTIONS

Throughout this study, let (Ω, F, {t}t≥) be a complete probability space with a filtration {Ft}t≥0. Moreover, |.| is the Euclidean norm in Rm and |ξ| is defined by Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching. Let ξ(t), t∈[-τ, 0 be F0 measurable and right-continuous and E||ξ||2<∞. Let w(t) be a one-dimensional Brownian motion defined on the probability space. Let w(t), r(t), t≥0, be a right-continuous Markov chain on the probability space taking values in a finite state space S = {1, 2,…, N} with the generator Γ = (γij)NxN given by:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

where, δ>0. Here γij≥0 is the transition rate from i to j if i≠j whil eImage for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching. The Markov chain r(t) is independent of the Brownian motion w(t). It is well known that almost every sample path of r(.) is a right-continuous step function with finite number of simple jumps in any finite subinterval of R+ = [0, +∞).

To analyze the Euler-Maruyama scheme as well as to simulate the approximate solution, the following lemma is useful (Mao and Yuan, 2006).

Lemma 1: Given Δ>0, let rΔk for k≥0. Then {rΔk, k = 1, 2,þ} is a discrete Markov chain with the one-step transition probability matrix:

P(Δ) = (Pij(Δ))NxN = eΔΓ

In this study, consider the scalar test equation with Markovian switching

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching
(2)

With initial data x0 = ξ∈C([-τ, 0]; R) and r(0) = r0∈S, where a(.), b(.), c(.), d(.)∈R, w(t) is a standard one-dimensional Brownian motion. The initial data ξ and i0 could be random, but the Markov property ensures that it is sufficient to consider only the case when both x0 and i0 are constants. It is known that the existence and uniqueness of the solutions are ensured under the local Lipschitz condition and the linear growth condition. From Mao and Yuan (2006), the following theorem is obvious.

Theorem 1: If for any i∈S, the following inequality:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching
(3)

holds. Then the solution of Eq. 2 is mean-square stable, that is:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching
(4)

SEMI-IMPLICIT MILSTEIN SCHEME

Now the adaptation of the semi-implicit Milstein method to Eq. 2 leads to a numerical scheme of the following form:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching
(5)

where 0≤α≤1, Δ>0 is a stepsize which satisfies τ = MΔ for some positive integer m and tn = nΔ, rΔn∈S. yn is an approximation to xn if tn≥0 then yn = ξ(tn). Moreover, Δwn = w(tn+1)-w(tn) are independent. yn is Ftn measurable at the mesh-point tn. Let I1 and I2 denote the two double integrals defined, respectively, by:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

The following lemma (Wang and Zhang, 2006) will be useful to the proof of the main result.

Lemma 2: The double integrals I1 and I2 satisfy EI1 = EI2 = E(I1I2) = 0:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

NUMERICAL STABILITY ANALYSIS

In this section the stability of the semi-implicit Milstein numerical method is given.

Definition 1: Under condition 3, a numerical method is said to be mean-square stable(MS--stable), if there exists a Δ0>0 such that the numerical solution sequence yn produced by this numerical scheme satisfies Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching, for every stepsize Δ∈(0, Δ0) with Δ = τ/m, where Δ0>0 dependents on a(.), b(.), c(.), d(.), m is an integer.

Definition 2: Under condition 3, a numerical method is said to be general mean-square stable (GMS--stable), if any application of the method to problem 2 generates numerical approximations yn which satisfy Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching, for every stepsize Δ = τ/m and an integer m.

As follows, the main theorem of this study is give:

Theorem 2: Assume that for any i∈S, the inequality (3) holds and:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

If L<0, then for every α∈[0, 1], the semi-implicit Milstein scheme is GMS-stable
If L≥0, then for every α∈(l, 1], the semi-implicit Milstein scheme is GMS-stable; for α∈[0, L], it is MS-stable and Δ, where Δ' = max{Δ1, Δ2}

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

and

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

Proof: To analyze the stability of the semi-implicit Milstein scheme, by Lemma 1, the generation of Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switchingoccurs before computing yn+1, then Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching is known. Since Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching∈s, for any i∈s, from (5), then

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

Note that Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switchingand Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching α,β∈R. Let. Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian SwitchingIt holds that:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

Where:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

Note that by (3) implies 1-αa(i)≠0 for any i∈S, then:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching
(7)

By recursive calculation, Yn→0(n→∞) if:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

which is equivalent to:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

If Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching then:

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

Since Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

It is obvious that if :

Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching

Thus Yn→0(n→∞) From (3), Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching , then if L<α≤1, then the semi-implicit Milstein method is GMS-stable, as a consequence, when L<0 and 0<α≤1, the method is GMS-stable and if 0<α≤1, then ,Image for - Preserving Mean-square Stability in the Simulation of Stochastic Differential Delay Equations with Markovian Switching thus the method is MS-stable. This proves the theorem.

CONCLUSION

This study is concerned with stability of the semi-implicit Milstein scheme of stochastic differential delay equations with Markovian switching. The GMS-stability and MS-stability of the semi-implicit Milstein method are proved. The obtained result shows that the numerical scheme reproduces the stability of the analytical solution.

ACKNOWLEDGMENT

The study is supported by the Research Fund for Wuhan Polytechnic University (2012Y16), the Fundamental Research Funds for the Central Universities (2722013JC080), the China Postdoctoral Science Foundation (2012M511615).

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