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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:
65C30Stochastic differential and integral equations
34K50Stochastic functional-differential equations
34F05ODE with randomness
65H10Systems of nonlinear equations (numerical methods)
65L06Multistep, Runge-Kutta, and extrapolation methods
65L20Stability and convergence of numerical methods for ODE
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References:
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