Introduction to the numerical analysis of stochastic delay differential equations.

*(English)*Zbl 0971.65004This paper concerns the numerical approximation of the strong solution of the ItĂ´ stochastic delay differential equation (SDDE)

$$dX\left(t\right)=f\left(X\right(t),X(t-\tau \left)\right)dt+g\left(X\right(t),X(t-\tau \left)\right)dW\left(t\right),\phantom{\rule{1.em}{0ex}}t\in [0,\tau ],$$

where $X\left(t\right)=\psi \left(t\right)$, $t\in [-\tau ,0]$ and $W\left(t\right)$ 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.

Reviewer: Melvin D.Lax (Long Beach)

##### 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 |