Exponential stability of θ-EM method for nonlinear stochastic Volterra integro-differential equations

Abstract

Mean square exponential stability of both exact solutions and the corresponding θ-EM method for stochastic Volterra integro-differential equations are investigated in this paper. For 12<θ≤1, we prove that both exact solutions and the corresponding θ-EM method for stochastic Volterra integro-differential equations are mean square exponentially stable under the Khasminskii-type conditions. If 0≤θ≤12, θ-EM method is mean square exponentially stable under the Khasminskii-type condition plus linear growth condition on f. By using Chebyshev inequality and Borel-Cantelli lemma, we can also prove that θ-EM method is almost surely exponentially stable. An example is provided to support our conclusions.