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Efficient Simulation for the Maximum of Infinite Horizon Discrete-Time Gaussian Processes

Part of: Limit theorems Probabilistic methods, simulation and stochastic differential equations Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Jose Blanchet  and
Chenxin Li
Jose Blanchet*
Affiliation:
Columbia University
Chenxin Li*
Affiliation:
Columbia University
*
∗Postal address: Department of Industrial Engineering and Operations Research, Columbia University, 321 S. W. Mudd, 500 West 120th Street, New York, NY 10027, USA.
∗Postal address: Department of Industrial Engineering and Operations Research, Columbia University, 321 S. W. Mudd, 500 West 120th Street, New York, NY 10027, USA.
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Abstract

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We consider the problem of estimating the probability that the maximum of a Gaussian process with negative mean and indexed by positive integers reaches a high level, say b. In great generality such a probability converges to 0 exponentially fast in a power of b. Under mild assumptions on the marginal distributions of the process and no assumption on the correlation structure, we develop an importance sampling procedure, called the target bridge sampler (TBS), which takes a polynomial (in b) number of function evaluations to achieve a small relative error. The procedure also yields samples of the underlying process conditioned on hitting b in finite time. In addition, we apply our method to the problem of estimating the tail of the maximum of a superposition of a large number, n, of independent Gaussian sources. In this situation TBS achieves a prescribed relative error with a bounded number of function evaluations as n ↗ ∞. A remarkable feature of TBS is that it is not based on exponential changes of measure. Our numerical experiments validate the performance indicated by our theoretical findings.

Keywords

Importance samplingrare-event simulationGaussian processlarge deviationsfractional Brownian noise

MSC classification

Secondary: 65C05: Monte Carlo methods 60G15: Gaussian processes 60F10: Large deviations
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 467 - 489
Copyright
Copyright © Applied Probability Trust 2011 

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