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The improved split-step backward Euler method for stochastic differential delay equations. (English) Zbl 1235.65010

The authors consider the numerical integration of stochastic differential delay equations

$dx\left(t\right)=f\left(x\left(t\right),x\left(t-\tau \left(t\right)\right)\right)dt+g\left(x\left(t\right),x\left(t-\tau \left(t\right)\right)\right)dw\left(t\right)$

with initial data $x\left(t\right)=\psi \left(t\right),\phantom{\rule{4pt}{0ex}}t\in \left[-\tau ,0\right]·$ Here $\tau \left(t\right)\ge 0,\phantom{\rule{4pt}{0ex}}-\tau :=inf\left\{t-\tau \left(t\right):t\ge 0\right\}$, $x$ is a $d$-dimensional vector, $w\left(t\right)$ is an $m$-dimensional Wiener process. For the equation, they introduce a new Euler method and prove its convergence in the mean-square sense. Further, the exponential mean-square stability of the proposed method is investigated.

##### MSC:
 65C30 Stochastic differential and integral equations 60H35 Computational methods for stochastic equations 60H10 Stochastic ordinary differential equations 65L20 Stability and convergence of numerical methods for ODE 34K50 Stochastic functional-differential equations 65L06 Multistep, Runge-Kutta, and extrapolation methods