Polynomial stability of exact solution and a numerical method for stochastic differential equations with time-dependent delay

Abstract

Polynomial stability of exact solution and modified truncated Euler-Maruyama method for stochastic differential equations with time-dependent delay are investigated in this paper. By using the well known discrete semimartingale convergence theorem, sufficient conditions are obtained for both bounded and unbounded delay δ to ensure the polynomial stability of the corresponding numerical approximation. Examples are presented to illustrate the conclusion.MSC 2010: 60H10, 65C30.there are few papers concerning about the polynomial stability of the numerical solution for the underlying stochastic differential equations with unbounded delay except [15]. Recently, Mao [12] introduced truncated EM method for stochastic differential equation without delay, and then he obtained sufficient conditions for the strong convergence rate of it in [13]. Motivated by these two works, we have introduced in [5] a new numerical simulation (which we called modified truncated Euler-Maruyama method) and obtained the strong convergence rate of it. Then we investigated p-th moment exponential stability of it in [6].In this paper, we will first extend modified truncated Euler-Maruyama method for stochastic differential equations to that of stochastic differential equations with time dependent delay (both bounded and unbounded cases), and then we will investigate the almost sure and mean square polynomial stability of the given modified truncated Euler-Maruyama method.Let (Ω, F , P ) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e. it is right continuous and F 0 contains all P-null sets). Let τ ≥ 0 be a constant and C ([−τ, 0] ; R n ) the space of all continuous functions from [−τ, 0] to R n with the norm φ = sup −τ ≤θ≤0 |φ (θ)|. Denote by C b F 0 ([−τ, 0]; R n ) the family of bounded, F 0 measurable, C ([−τ, 0] ; R n )-valued random variables. Let B (t) be a d-dimensional standard Brownian motion.Consider the following stochastic differential delay equations:with the initial valuewhere δ(t) ∈ C 1 (R + , R + ) such that δ(0) = τ , E||ξ|| 2 < ∞, moreover, f : R n ×R n ×[0, +∞) → R n and g : R n × R n × [0, +∞) → R n ⊗ R d are Borel measurable vector and matrix valued functions, respectively.Notice that stochastic pantograph equation is a special case of the above stochastic delay differential equation (1.1) with unbounded memory (i.e. δ(t) = t − qt, 0 < q < 1 and x 0 = x(0) ∈ R n is a F 0 measurable random variable).We always assume that f (0, 0, t) ≡ 0, g(0, 0, t) ≡ 0, which implies that X ≡ 0 is the trivial solution of equation (1.1). And we assume thatThis implies that t − δ(t) is strictly increasing on [0, ∞).We impose two standing hypotheses on f and g in this paper.Assumption 1.1 The coefficients f and g satisfy local Lipschitz condition for any fixed t > 0, that is, for each R and t there is L R,t > 0 such that |f (x, y, t) − f (x ′ , y ′ , t)| ∨ |g(x, y, t) − g(x ′ , y ′ , t)| ≤ L R,t (|x − x ′ | + |y − y ′ |) (1.3)for all |x| ∨ |x ′ | ∨ |y| ∨ |y ′ | ≤ R.

The truncated EM method for stochastic differential delay equations with variable delay

“…Moreover, Song et al [18] achieved a better convergence order than [14] and [17] by adopting the truncation techniques from [19] for such SDDEs. The applications of the partially truncated and modified truncated EM methods for SDDEs can be found in [20,21]. Other explicit numerical methods for superlinear SDDEs, say tamed EM, balanced EM, truncated Milstein, projected EM, are discussed in [22,23,24,25,26,27].…”

Equivalence of pth moment stability between stochastic differential delay equations and their numerical methods

“…Secondly, the requirement on the step size of the method is significantly released compared with the existing works, which is a stand-alone interesting result. It should be mentioned that many other interesting numerical methods have been proposed for SDDEs, for example [1,2,3,4,8,9,11,10,17,18,23] and the references therein.…”