A confidence interval for Monte Carlo methods with an application to simulation of obliquely reflecting Brownian motion

This paper deals with the estimate of errors introduced by finite sampling in Monte Carlo evaluation of functionals of stochastic processes. To this end we introduce a metric d over the space of probability measures which induces a topology finer than the weak topology. For any two measures μ, v, this metric allows to bound /vb〈μ,f〉 - 〈v,f〉/vb, uniformly over a large class of C1-functions f, by a quantity which can be computed by a finite number of calculations. In the case v = μn, the empirical distribution of order n of μ, we can compute the minimum sample size that will ensure that this quantity will be smaller than any given ε, at any chosen confidence level. As an application we control the rate of convergence of an approximating scheme for obliquely reflecting Brownian motion on a half-plane by a Monte Carlo evaluation of two significant functionals on the path space.