STANFORD UNIVERSITY -- SCCM SEMINARS
CS 531
Winter 2002

Monday, January 28, 2002
4:15 - 5:15pm
Gates B12

Dr Vince Fernando
Computational Engines Ltd, Oxford, England
vince_fernando@lycos.co.uk

Simulation of the Ornstein-Uhlenbeck and Wiener Processes

The Ornstein-Uhlenbeck process (OUP) is one of the fundamental equations in physics and is used to describe diffusion with a drift. Apart from applications in many branches of physics (e.g. astrophysics, geophysics, lasers, Johnson noise), OUP is fundamental in modeling financial systems and neurons. Statistically, OUP is a continuous-time Markov process and is described by a linear stochastic differential equation, which is driven by a Wiener process. It is possible to simulate the OUP by numerical integration. However, accurate simulations of the sampled process, without truncation errors, can be done by studying the covariance matrix and deriving a discrete Markov process for the sampled values. The primary aim of this research is to study the covariance matrix and use the structural properties of the covariance for efficient simulation.

The covariance matrix associated with the Wiener process is also somewhat embedded in the OUP covariance. The analysis is based on the Cholesky factorization of the covariance matrix and the inverse of this Cholesky factor is lower bidiagonal even when the sampling intervals are not uniform. It is possible to derive simple explicit formulae to compute the elements of this bidiagonal. This leads to forward simulation of the OUP, which can be considered as an initial value problem with deterministically or statistically known initial values. The dominant modes of the system are given by the inverse of the bidiagonal and hence reduced-order simulations may be possible by neglecting modes corresponding to large singular values.

Similarly, reverse Cholesky factorization gives backward simulation, which is equivalent to a final value problem; the inverse of the reverse Cholesky factor is then upper bidiagonal. It is possible to use coarse time steps initially and then refine the sample paths between these coarse time steps. This approach is similar to solving a two-point boundary value problem between two adjacent coarse time steps, and in stochastic theory literature they are known as bridges. Bridges for the OUP can be obtained by burn-at-both-ends (BABE) factorization of the inverse of the covariance. The canonical bidiagonal structure is common to the AR(1) process (which can be considered as a uniformly sampled OUP) and the Wiener process. Similar results can also be derived for more complex processes defined by random Gaussian fields and Brownian sheets.