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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 $$dx(t)= f(x(t), x(t-\tau),t)\,dt+ g(x(t), x(t-\tau), t)\,dw(t),\quad t\ge 0.$$ 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 $f$.

65C30Stochastic differential and integral equations
65C99Probabilistic methods, simulation and stochastic differential equations (numerical analysis)
Full Text: DOI
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