Numerical methods for some nonlinear stochastic differential equations

摘要

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.