## The Annals of Applied Probability

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

### Superextremal Processes, Max-Stability and Dynamic Continuous Choice

Sidney I. Resnick and Rishin Roy

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177004972

**Digital Object Identifier**

doi:10.1214/aoap/1177004972

**Mathematical Reviews number (MathSciNet)**

MR1284986

**Zentralblatt MATH identifier**

0809.60064

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G70: Extreme value theory; extremal processes

Secondary: 60G55: Point processes

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aoap/1177004972