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.
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):
(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
where, δ>0. Here γij≥0 is the transition rate from i to j if i≠j whil e
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
(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:
(3) |
holds. Then the solution of Eq. 2 is mean-square stable, that is:
(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:
(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:
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:
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
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
As follows, the main theorem of this study is give:
Theorem 2: Assume that for any i∈S, the inequality (3) holds and:
• | 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} |
and
Proof: To analyze the stability of the semi-implicit Milstein scheme, by Lemma 1, the generation of
Note that
Where:
Note that by (3) implies 1-αa(i)≠0 for any i∈S, then:
(7) |
By recursive calculation, Yn→0(n→∞) if:
which is equivalent to:
If
Since
It is obvious that if :
Thus Yn→0(n→∞) From (3),
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).