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Almost sure exponential stability of numerical solutions for stochastic delay differential equations. (English) Zbl 1193.65009

A theorem is proved that gives sufficient conditions for almost sure exponential stability (ASES) of Euler-Maruyama method numerical solutions of the \(n\)-dimensional nonlinear stochastic delay differential equation \[ dx(t)= f(x(t), x(t-\tau),t)\,dt+ g(x(t), x(t-\tau), t)\,dw(t),\quad t\geq 0. \] A counterexample is presented to show that without the linear growth condition on \(f\) of the theorem, ASES may be lost. Then for the backward Euler-Maruyama method ASES is proved when a one-sided Lipschitz condition on \(f\) in \(x\) replaces the linear growth condition on \(f\).

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
65C99 Probabilistic methods, stochastic differential equations

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References:

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