Superextremal Processes, Max-Stability and Dynamic Continuous Choice

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.