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Introduction to the numerical analysis of stochastic delay differential equations. (English) Zbl 0971.65004
This paper concerns the numerical approximation of the strong solution of the Itô stochastic delay differential equation (SDDE) $$dX(t)=f(X(t),X(t-\tau))dt+g(X(t),X(t-\tau))dW(t),\quad t\in[0,\tau],$$ where $X(t) =\psi(t)$, $t\in [-\tau,0]$ and $W(t)$ is a Wiener process. A theorem is proved establishing conditions for convergence, in the mean-square sense, of approximate solutions obtained from explicit single-step methods. Then a SDDE version of the Euler-Maruyama method is presented and found to have order of convergence 1. The paper concludes with several figures illustrating numerical results obtained when this method is applied to an example.

##### MSC:
 65C30 Stochastic differential and integral equations 34K50 Stochastic functional-differential equations 34F05 ODE with randomness 65H10 Systems of nonlinear equations (numerical methods) 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L20 Stability and convergence of numerical methods for ODE
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