The use of importance sampling in the solution of stochastic differential equations. (Q1432641)

The authors consider diffusion processes given by a system of stochastic differential equations of the form \[ dy(t) = a(y(t))dt +b(y(t))dW(t),\tag{1} \] for \(0\leq t\leq T\) and with the initial value \(y(0)=y_0\). Denote by \(p(T,t,x)\) the probability that the stochastic process \(y(.)\) does not reach the boundary \(\Gamma\) of a region \(\Omega\) in a time between \(t\) and \(T\), provided that \(y\) started at \(x\in \Omega\). Then the problem investigated in this article is the numerical estimation of \(p(T,0,y_0)\). The stochastic differential equation (1) is approximated by the Euler-Maruyama method. The authors discuss several estimators, in particular a discrete and a continuous one and a weighted estimator, given by importance sampling applied to the discrete estimator.