Variance reduction for Monte Carlo simulation of stochastic environmental models

Abstract

To determine the probability of exceedence Monte Carlo simulation of stochastic models is often used. Mathematically this requires the evaluation of an expectation of some function of a solution of a stochastic model. This can be reformulated as a Kolmogorov final value problem. It can thus be calculated numerically by either solving a deterministic partial differential equation (Kolmogorov's Backwards equations) or by simulating a large number of trajectories of the stochastic differential equation. Here we discuss a composite method of variance reduced Monte Carlo simulation. The variance reduction is obtained by the Girsanov transformation to modify the stochastic model by a correction term that is obtained from an approximate solution of the partial differential equation computed by a classical numerical method. The composite method is more efficient than either the standard Monte Carlo or the classical numerical method. The approach is applied to estimate the probability of exceedence in a model for biochemical-oxygen demand.