Stability of analytical and numerical solutions of nonlinear stochastic delay differential equations
For researchers working on numerical methods for stochastic delay differential equations, this work provides a unified theoretical framework for stability analysis without requiring Lyapunov functionals, but the results are incremental as they extend known concepts to a broader class of SDDEs.
This paper derives sufficient conditions for stability, contractivity, and asymptotic contractivity in mean square of solutions for nonlinear stochastic delay differential equations (SDDEs) with constant and variable delays, and shows that the backward Euler method preserves these properties.
This paper concerns the stability of analytical and numerical solutions of nonlinear stochastic delay differential equations (SDDEs). We derive sufficient conditions for the stability, contractivity and asymptotic contractivity in mean square of the solutions for nonlinear SDDEs. The results provide a unified theoretical treatment for SDDEs with constant delay and variable delay (including bounded and unbounded variable delays). Then the stability, contractivity and asymptotic contractivity in mean square are investigated for the backward Euler method. It is shown that the backward Euler method preserves the properties of the underlying SDDEs. The main results obtained in this work are different from those of Razumikhin-type theorems. Indeed, our results hold without the necessity of constructing of finding an appropriate Lyapunov functional.