Theta schemes for SDDEs with non-globally Lipschitz continuous coefficients

Abstract

This paper establishes the boundedness, convergence and stability of the two classes of theta schemes, namely split-step theta (SST) scheme and stochastic linear theta (SLT) scheme, for stochastic differential delay equations (SDDEs) with non-globally Lipschitz continuous coefficients. When the drift $f(x,y)$ satisfies one-sided Lipschitz condition with respect to the present state $x$ and the diffusion $g(x,y)$ obeys the global Lipschitz condition with respect to the present term $x$, but the delay terms $y$ in the drift and diffusion may be highly nonlinear, this paper first examines the strong convergence rates of the theta schemes for SDDEs. It is also proved that the two classes of theta schemes for $\theta \in (1/2,1]$ converge strongly to the exact solution with the order $1/2$ but for $\theta \in [0,1/2]$ the linear growth condition on drift $f(x,y)$ in $x$ is needed for the strong convergence rates. The exponential mean square stability of the theta schemes with $\theta \in (1/2,1]$ is also investigated for highly nonlinear SDDEs.