Analysis on Mean-Square and Almost Sure Exponential Stability of Numerical Method for Stochastic Differential Equations with Constant Delay
In this paper we consider the Milstein method to investigate the mean-square and almost sure exponential stability for nonlinear stochastic delay differential equations (SDDEs). It is shown that the Milstein method reproduces the mean-square and almost sure exponential stability under an additional linear growth condition. Also in this regard, we employ the discrete semimartingale convergence theorem for almost sure exponential stability. In the numerical section we present the stability results by simulating the method on kind of equations.
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