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Zbl 0971.65004
Buckwar, Evelyn
Introduction to the numerical analysis of stochastic delay differential equations.
(English)
[J] J. Comput. Appl. Math. 125, No.1-2, 297-307 (2000). ISSN 0377-0427

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.
[Melvin D.Lax (Long Beach)]
MSC 2000:
*65C30 Stochastic differential and integral equations
34K50 Stochastic delay equations
34F05 ODE with randomness
65H10 Systems of nonlinear equations (numerical methods)
65L06 Multistep, Runge-Kutta, and extrapolation methods
65L20 Stability of numerical methods for ODE

Keywords: strong solution; Ito stochastic delay differential equation; convergence; explicit single-step methods; Euler-Maruyama method; numerical results

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