Generating Quasi-Random Paths for Stochastic Processes

The need to numerically simulate stochastic processes arises in many elds. Frequently this is done by discretizing the process into small time steps and applying pseudo-random sequences to simulate the randomness. This paper address the question of how to use quasi-Monte Carlo methods to improve this simulation. Special techniques must be applied to avoid the problem of high dimensionality which arises when a large number of time steps are required. Two such techniques, the generalized Brown-ian bridge and particle reordering, are described here. These methods are applied to a problem from nance, the valuation of a 30 year bond with monthly coupon payments assuming a mean reverting stochastic interest rate. When expressed as an integral, this problem is nominally 360 dimensional. The analysis of the integrand presented here explains the eeectiveness of the quasi-random sequences on this high dimensional problem and suggests methods of variance reduction which can be used in conjunction with the quasi-random sequences.