Stability Analysis of Numerical Schemes for Stochastic Differential Equations
SIAM Journal on Numerical Analysis
Introduction to the numerical analysis of stochastic delay differential equations
Journal of Computational and Applied Mathematics - Special issue on numerical anaylsis 2000 Vol. VI: Ordinary differential equations and integral equations
Mean-Square and Asymptotic Stability of the Stochastic Theta Method
SIAM Journal on Numerical Analysis
Numerical solutions of stochastic differential delay equations under local Lipschitz condition
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
Delay-dependent stability analysis of numerical methods for stochastic delay differential equations
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
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Our aim is to study under what conditions the exact and numerical solution (based on equidistant nonrandom partitions of integration time-intervals) to a stochastic differential delay equation (SDDE) share the property of mean-square exponential stability. Our approach is trying to avoid the use of Lyapunov functions or functionals. We show that under a global Lipschitz assumption an SDDE is exponentially stable in mean square if and only if for some sufficiently small stepsize @D the Euler-Maruyama (EM) method is exponentially stable in mean square. We then replace the global Lipschitz condition with a finite-time convergence condition and establish the same ''if and only if'' result. The important feature of this result is that it transfers the asymptotic problem into a finite-time convergence problem. Replacing the exact and EM numerical solution with stochastic processes, we also discuss whether a family of stochastic processes share the stability property. This new approach allows us to discuss (i) whether a family of SDDEs share the stability property, and (ii) whether an SDDE with variable time lag shares stability property with the modified EM method. As another application of this general approach we consider a linear SDDE with variable time lag and establish an ''if and only if'' result. It should also be mentioned that the questions addressed, results proved, as well as style of analysis borrow heavily from [14] but the computations involved to cope with time delay are nontrivial.