The Annals of Applied Probability

Superextremal Processes, Max-Stability and Dynamic Continuous Choice

Sidney I. Resnick and Rishin Roy

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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.

Article information

Ann. Appl. Probab., Volume 4, Number 3 (1994), 791-811.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G55: Point processes

Choice theory superextremal processes extreme value theory Poisson process max-stability


Resnick, Sidney I.; Roy, Rishin. Superextremal Processes, Max-Stability and Dynamic Continuous Choice. Ann. Appl. Probab. 4 (1994), no. 3, 791--811. doi:10.1214/aoap/1177004972.

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