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A dichotomy for sampling barrier-crossing events of random walks with regularly varying tails

Part of: Stochastic processes Algorithms - Computer Science Operations research and management science

Published online by Cambridge University Press:  30 November 2017

A. B. Dieker  and
Guido R. Lagos
A. B. Dieker*
Affiliation:
Columbia University
Guido R. Lagos*
Affiliation:
Universidad de Chile
*
* Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA.
** Postal address: Center for Mathematical Modeling, Universidad de Chile, Beauchef 851, Torre Norte oficina 705, Santiago, RM 8370456, Chile. Email address: [email protected]
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Abstract

We study how to sample paths of a random walk up to the first time it crosses a fixed barrier, in the setting where the step sizes are independent and identically distributed with negative mean and have a regularly varying right tail. We introduce a desirable property for a change of measure to be suitable for exact simulation. We study whether the change of measure of Blanchet and Glynn (2008) satisfies this property and show that it does so if and only if the tail index α of the right tail lies in the interval (1, 3/2).

Keywords

Conditional samplingperfect samplingefficiencyrandom walkheavy tailregularly varying tailchange of measurelikelihood ratio

MSC classification

Primary: 68U20: Simulation 60G50: Sums of independent random variables; random walks 68W40: Analysis of algorithms
Secondary: 90B99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1213 - 1232
Copyright
Copyright © Applied Probability Trust 2017 

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