Generating Quasi-Random Paths for Stochastic Processes

The need to simulate stochastic processes numerically arises in many fields. Frequently this is done by discretizing the process into small time steps and applying pseudorandom sequences to simulate the randomness. This paper addresses the question of how to use quasi-Monte Carlo methods to improve...

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Bibliographic Details
Published in:SIAM review 1998, Vol.40 (4), p.765-788
Main Author: Morokoff, William J.
Format: Article
Language:English
Subjects:
Applied sciences
Brownian bridge
Brownian motion
Exact sciences and technology
Finance
Hessian matrices
Integrands
Interest rates
Mathematical functions
Mathematics
Matrices
Monte Carlo simulation
Numerical analysis
Numerical analysis. Scientific computation
Numerical simulation
Operational research and scientific management
Operational research. Management science
Random walk
Sciences and techniques of general use
Stochastic models
Stochastic processes
Taylor series
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Summary:The need to simulate stochastic processes numerically arises in many fields. Frequently this is done by discretizing the process into small time steps and applying pseudorandom sequences to simulate the randomness. This paper addresses 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 is required. Two such techniques, the generalized Brownian bridge and particle reordering, are described here. These methods are applied to a problem from finance, 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 effectiveness 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.
ISSN:0036-1445
1095-7200
DOI:10.1137/s0036144597317959