Almost sure exponential stability of stochastic differential delay equations

This paper mainly studies whether the almost sure exponential stability of stochastic differential delay equations (SDDEs) is shared with that of the stochastic theta method. We show that under the global Lipschitz condition the SDDE is pth moment exponentially stable (for p 2 (0; 1)) if and only if the stochastic theta method of the SDDE is pth moment exponentially stable and pth moment exponential stability of the SDDE or the stochastic theta method implies the almost sure exponential stability of the SDDE or the stochastic theta method, respectively. We then replace the global Lipschitz condition with a finite-time convergence condition and establish the same results. Hence, our new theory enables us to consider the almost sure exponential stability of the SDDEs using the stochastic theta method, instead of the method of Lyapunov functions. That is, we can now perform careful numerical simulations using the stochastic theta method with a sufficiently small step size ?t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ? (0,1), we can then deduce that the underlying SDDE is almost sure exponentially stable. Our new theory also enables us to show the pth moment exponential stability of the stochastic theta method to reproduce the almost sure exponential stability of the SDDEs.

A revisit of stochastic theta method with some improvements

Considering the stochastic theta method, in this work we discuss modified solvers for finding the solution of It? stochastic differential equations in the strong sense. It is observed that the proposed variations are implicit with better stability behaviors. Finally, the derived schemes are tested numerically to confirm the efficiency.