Almost Sure Exponential Stability Of One-Leg Theta-methods Of Stochastic Differential Delay Equations

This paper mainly studies the p th moment exponential stability and the almost sure exponential stability of analytic solutions and numerical solutions of stochastic delay differential functions(SDDEs). We usually use Lyapunov functions to study stabilities of SDDEs. But it is not always successful to find appropriate Lyapunov functions for all SDDEs. In the absence of an appropriate function, we can carry out careful numerical simulations using a numerical method with a “small” step size ?t. One of the important subjects of many research fields of numerical methods is to inspect their ability to preserve the stable behavior of solutions of original systems.Firstly, we study the p th moment exponential stability of stochastic delay differential equations. The equivalence relation between the p th moment exponential stability of analytic solutions and numerical solutions is shown that SDDEs are p th moment exponentially stable(for p ∈(0,1)) if and only if their stochastic one-leg theta methods are p th moment exponentially stable for sufficiently small step sizes under the global-Lipschitz condition and the liner-growth condition.Secondly, we investigate the almost sure exponential stability of stochastic delay differential equations. It is shown that the almost sure exponential stability of SDDEs or their stochastic one-leg theta methods can be reproduced from the p th moment exponential stability of SDDEs or their stochastic one-leg theta methods under the same conditions.Finally, this paper states that we can use stochastic one-leg theta methods to study stabilities of the SDDEs instead of Lyapunov functions. If the stochastic one-leg theta method is p th moment exponentially stable for a sufficiently small p ∈(0,1), we can then infer that the underlying SDDE isp th moment exponentially stable and almost sure exponentially stable.