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.
Almost sure exponential stability of stochastic differential delay equations
