Monte Carlo Simulation of Boundary Crossing Probabilities for a Brownian Motion and Curved Boundaries

摘要

We are concerned with the probability that a standard Brownian motion W(t) crosses a curved boundary c(t) on a finite interval [0, T]. Let this probability be denoted by Q(c(t); T). Due to recent advances in research a new way of estimating Q(c(t); T) seems feasible: Monte Carlo Simulation. Wang and Pötzelberger (1997) derived an explicit formula for the boundary crossing probability of piecewise linear functions which has the form of an expectation. Based on this formula we proceed as follows: First we approximate the general boundary c(t) by a piecewise linear function cm(t) on a uniform partition. Then we simulate Brownian sample paths in order to evaluate the expectation in the formula of the authors for cm(t). The bias resulting when estimating Q(c_m(t); T) rather than Q(c(t); T) can be bounded by a formula of Borovkov and Novikov (2005). Here the standard deviation - or the variance respectively - is the main limiting factor when increasing the accuracy. The main goal of this dissertation is to find and evaluate variance reducing techniques in order to enhance the quality of the Monte Carlo estimator for Q(c(t); T). Among the techniques we discuss are: Antithetic Sampling, Stratified Sampling, Importance Sampling, Control Variates, Transforming the original problem. We analyze each of these techniques thoroughly from a theoretical point of view. Further, we test each technique empirically through simulation experiments on several carefully chosen boundaries. In order to asses our results we set them in relation to a previously established benchmark. As a result of this dissertation we derive some very potent techniques that yield a substantial improvement in terms of accuracy. Further, we provide a detailed record of our simulation experiments. (author's abstract)