Introduction to the numerical analysis of stochastic delay differential equations (Q1841963)

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.