Euler-Maruyama Numerical solution of some stochastic functional differential equations

Abstract

In this paper we study the numerical solutions  of the stochastic functional differential equations of the following form $du(x,t) = f(x,t,u_t)dt + g(x,t,u_t)dB(t),~ t>0$ with initial data $u(x,0)= u_0(x)=\xi \in L^p_{F_0}([-\tau,0];R^n)$

Here $x \in R^n$ ($R^n$ is the $\nu$-dimenional Euclidean space),

$f: C([-\tau,0]; R^n )\times R^{\nu + 1} \rightarrow R^n$

$g: C([-\tau,0];R^n)\times R^{\nu + 1}\rightarrow R^{n \times m } u(x,t)\in R^n$ for each $t$,

$u_t = {u(x,t+ \theta ):-\tau\leq\theta\leq 0}\in C([-\tau,0];R^n)$ and $B(t)$ is an m-dimensional Brownian motion.