Almost sure exponential stability of stochastic differential delay equation

Abstract

This paper is mainly concerned with 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
exponential stable (for $p \in (0,1)$) if and only if the
stochastic theta method is $pth$ moment exponential stability of the
SDDE or the stochastic theta method and $pth$ moment exponential stability of the SDDE or the stochastic theta method implies the almost sure
exponentially stable 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 enable us to study the the almost sure exponentially stable of the SDDEs using the stochastic theta method, instead of the method of Lyapunov functions. That is, we can now carry out careful numerical simulations using the
stochastic theta method with a sufficiently small step size $\Delta
t$. If the stochastic theta method is $pth$ moment exponential stable
for a sufficiently small $p \in (0,1)$, we can then infer that the
underlying SDDE is almost sure exponentially stable. Our new theory
also enables us to show the stability of the stochastic theta method
to reproduce the almost sure exponentially stability of the SDDEs.