Convergence and stability of exponential integrators for semi-linear stochastic variable delay integro-differential equations

In this paper, the numerical method for semi-linear stochastic variable delay integro-differential equations is studied. The stability of analytic solutions of semi-linear stochastic variable delay integro-differential equations are studied first, some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential integrators for semi-linear stochastic variable delay integro-differential equations are constructed, the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with the strong order 1/2 and the exponential Euler method can keep the mean-square exponential stability of the analytical solution under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.