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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

$dx\left(t\right)=f\left(x\left(t\right),x\left(t-\tau \right),t\right)\phantom{\rule{0.166667em}{0ex}}dt+g\left(x\left(t\right),x\left(t-\tau \right),t\right)\phantom{\rule{0.166667em}{0ex}}dw\left(t\right),\phantom{\rule{1.em}{0ex}}t\ge 0·$

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 $f$.

##### MSC:
 65C30 Stochastic differential and integral equations 65C99 Probabilistic methods, simulation and stochastic differential equations (numerical analysis)
RODAS
##### References:
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