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Super-extremal processes and the argmax process

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Sidney I. Resnick  and
Rishin Roy
Sidney I. Resnick*
Affiliation:
Cornell University
Rishin Roy*
Affiliation:
University of Toronto
*
∗Postal address: School of Operations Research and Industrial Engineering, ETC Building, Cornell University, Ithaca, NY 14853, USA.
∗∗Present address: Paribas Corporation, Equitable Tower, 787 7th Avenue, New York, NY 10019, USA.
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Abstract

In this paper, we develop the probabilistic foundations of the dynamic continuous choice problem. The underlying choice set is a compact metric space E such as the unit interval or the unit square. At each time point t, utilities for alternatives are given by a random function . To achieve a model of dynamic continuous choice, the theory of classical vector-valued extremal processes is extended to super-extremal processesY= {Yt, t > 0}. At any t > 0, Yt is a random upper semicontinuous function on a locally compact, separable, metric space E. General path properties of Y are discussed and it is shown that Y is Markov with state-space US(E). For each t > 0, Yt is associated.

For a compact metric E, we consider the argmax process M = {Mt, t > 0}, where . In the dynamic continuous choice application, the argmax process M represents the evolution of the set of random utility maximizing alternatives. M is a closed set-valued random process, and its path properties are investigated.

Keywords

DYNAMIC CONTINUOUS CHOICERANDOM CLOSED SETRANDOM UPPER SEMICONTINUOUS FUNCTIONSUPER-EXTREMAL PROCESSPOISSON PROCESSCLOSED SET-VALUED PROCESS

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G55: Point processes
Type
Research Article
Information
Journal of Applied Probability , Volume 31 , Issue 4 , December 1994 , pp. 958 - 978
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

Partially supported by NSF Grant DMS 9100027.

Partially supported by the Natural Sciences and Engineering Research Council of Canada.

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