Asymptotic Stability Analysis of a Stochastic Volterra Integro-differential Equation with Fading Memory

We investigate the long term behavior of solutions to a stochastic Volterra integro-differential equation with a fading memory; the fading memory is represented by using a decaying exponential convolution kernel. We give sufficient conditions for asymptotic mean square stability of the solution. In a similar spirit, we investigate the long term behavior of solutions to discrete analogues of the above continuous problem; our discrete analogues are based on the EulerMaruyama scheme for stochastic differential equations and θ-methods for approximating the integral term. We give necessary and sufficient conditions for asymptotic mean square stability of the trivial solution, obtaining our results by means of the general method of Lyapunov functionals construction. We focus also on the geometric interpretations of our findings, such as the sizes of stability regions. This enables us to make some conclusions with regards to choosing an appropriate θ-method for obtaining numerical approximations.