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August, 1994 Superextremal Processes, Max-Stability and Dynamic Continuous Choice
Sidney I. Resnick, Rishin Roy
Ann. Appl. Probab. 4(3): 791-811 (August, 1994). DOI: 10.1214/aoap/1177004972


A general framework in an ordinal utility setting for the analysis of dynamic choice from a continuum of alternatives $E$ is proposed. The model is based on the theory of random utility maximization in continuous time. We work with superextremal processes $\mathbf{Y} = \{\mathbf{Y}_t, t \in (0,\infty)\}$, where $\mathbf{Y}_t = \{Y_t(\tau),\tau \in E\}$ is a random element of the space of upper semicontinuous functions on a compact metric space $E$. Here $Y_t(\tau)$ represents the utility at time $t$ for alternative $\tau \in E$. The choice process $\mathbf{M} = \{M_t, t \in (0,\infty)\}$, is studied, where $M_t$ is the set of utility maximizing alternatives at time $t$, that is, $M_t$ is the set of $\tau \in E$ at which the sample paths of $\mathbf{Y}_t$ on $E$ achieve their maximum. Independence properties of $\mathbf{Y}$ and $\mathbf{M}$ are developed, and general conditions for $\mathbf{M}$ to have the Markov property are described. An example of such conditions is that $\mathbf{Y}$ have max-stable marginals.


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Sidney I. Resnick. Rishin Roy. "Superextremal Processes, Max-Stability and Dynamic Continuous Choice." Ann. Appl. Probab. 4 (3) 791 - 811, August, 1994.


Published: August, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0809.60064
MathSciNet: MR1284986
Digital Object Identifier: 10.1214/aoap/1177004972

Primary: 60G70
Secondary: 60G55

Keywords: Choice theory , Extreme value theory , max-stability , Poisson process , superextremal processes

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.4 • No. 3 • August, 1994
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