θ-Maruyama methods for nonlinear stochastic differential delay equations

Abstract

Nonlinear stochastic differential delay equations Variable lag θ-Maruyama methods Strong convergence Exponential mean-square stability Mean-square stabilityIn this paper, mean-square convergence and mean-square stability of θ -Maruyama methods are studied for nonlinear stochastic differential delay equations (SDDEs) with variable lag. Under global Lipschitz conditions, the methods are proved to be mean-square convergent with order 1 2 , and exponential mean-square stability of SDDEs implies that of the methods for sufficiently small step size h > 0. Further, the exponential mean-square stability properties of SDDEs and those of numerical methods are investigated under some non-global Lipschitz conditions on the drift term. It is shown in this setting that the θ -Maruyama method with θ = 1 can preserve the exponential mean-square stability for any step size. Additionally, the θ -Maruyama method with 1 2 ≤ θ ≤ 1 is asymptotically mean-square stable for any step size, provided that the underlying system with constant lag is exponentially mean-square stable. Applications of this work to some special problem classes show that the results are deeper or sharper than those in the literature. Finally, numerical experiments are included to demonstrate the obtained theoretical results.

A General Class of One-Step Approximation for Index-1 Stochastic Delay-Differential-Algebraic Equations

“…where τ = lh with integer l. It has been clarified in the proof of Theorem 2.1 that functionsf andg satisfy the uniform Lipschitz condition (2.12) and linear growth condition (2.13). Thus, the proof of Theorem 2 in [18] shows that method (4.11) is mean square consistent, i.e., the local error of method (4.11)…”

“…18b) x Following the skills of mean square consistency analysis of θ-Maruyama methods [18], we can prove thatδ k satisfy estimates (4.13) and (4.14) . Therefore, split-step θ-methods (…”

“…Employing semi-implicit Euler method for SDDEs (cf. [18]) to (4.8) we have the following discretization schemẽ…”

“…To our knowledge, for index-1 SDDAEs, Xiao and Zhang [21,22] derived some existence and uniqueness results and the Euler-Maruyama methods. Whereas, most of the existing stochastic numerical methods are only for stochastic delay differential equations (SDDEs) or stochastic differential equations (SDEs) without algebraic constrain, see, e.g., [4,5,[12][13][14]18] and their references.…”

Projected Euler-Maruyama method for stochastic delay differential equations under a global monotonicity condition

“…In 2002, Mao [6] gave the Khasminskii-type condition for SDDEs where linear growth condition was no longer necessary, and global existence and uniqueness of the solution was proved.Most of SDDEs can not be solved analytically, so numerical calculation is particularly necessary. In the past two decades, a number of numerical methods were investigated under Lipschitz and linear growth condition (see [7][8][9][10][11][12][13][14][15][16][17][18] and references therein). Limited work has been done in SDDEs whose coefficients do not satisfy the linear growth condition, and this issue received attention only recently.…”