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Processes with associated increments

Part of: Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Paul Glasserman
Paul Glasserman*
Affiliation:
Columbia University
*
∗Postal address: 403 Uris Hall, Columbia University, New York, NY 10027, USA. E-mail address: fapglass@cugsbvm.gsb.columbia.edu
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Abstract

We derive conditions under which the increments of a vector process are associated — i.e. under which all pairs of increasing functions of the increments are positively correlated. The process itself is associated if it is generated by a family of associated and monotone kernels. We show that the increments are associated if the kernels are associated and, in a suitable sense, convex. In the Markov case, we note a connection between associated increments and temporal stochastic convexity.

Our analysis is motivated by a question in variance reduction: assuming that a normalized process and its normalized compensator converge to the same value, which is the better estimator of that limit? Under some additional hypotheses we show that, for processes with conditionally associated increments, the compensator has smaller asymptotic variance.

Keywords

ASSOCIATED MEASURES AND KERNELSFKG INEQUALITIESSTOCHASTIC CONVEXITYCONVEX ORDERVARIANCE REDUCTION

MSC classification

Secondary: 65C05: Monte Carlo methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 313 - 333
Copyright
Copyright © Applied Probability Trust 1992 

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