Numerical methods for some nonlinear stochastic differential equations

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In this paper we study the numerical solutions of the stochastic differential equations of the formdu(x,t)=f(x,t,u)dt+g(x,t,u)dW(t)+∑∣q∣⩽2mAq(x,t)Dqu(x,t)dt,where 0⩽t⩽T,x∈Rν (Rν is the ν-dimensional Euclidean space).Here u∈Rn, W(t) is an n-dimensional Brownian motion,f:Rn+ν+1→Rn,g:Rn+ν+1→Rn×n,andAq:Rν×[0,T]→Rn×n,where (Aq,∣q∣⩽2m) is a family of square matrices whose elements are sufficiently smooth functions on Rν×[0,T] and Dq=D1q1⋯Dνqν, Di=∂∂xi.