Adaptive Monte Carlo algorithms for stopped diffusion

We present adaptive algorithms for weak approximation of stopped diffusion using the Monte Carlo Euler method. The goal is to compute an expected value E[g(X(τ), τ)] of a given function g depending on the solution X of an Itô stochastic differential equation and on the first exit time τ from a given domain. The adaptive algorithms are based on an extension of an error expansion with computable leading order term, for the approximation of E[g(X(T))] with a fixed final time T > 0 and diffusion processes X in ℝ^d, introduced in [17] using stochastic flows and dual backward solutions. The main steps in the extension to stopped diffusion processes are to use a conditional probability to estimate the first exit time error and introduce difference quotients to approximate the initial data of the dual solutions. Numerical results show that the adaptive algorithms achieve the time discretization error of order N^-1 with N adaptive time steps, while the error is of order N^-1/2 for a method with N uniform time steps.