Superextremal Processes, Max-Stability and Dynamic Continuous Choice

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 Y={Yt,t.(0,.)}, where Yt={Yt(.),..E} is a random element of the space of upper semicontinuous functions on a compact metric space E. Here Yt(.) represents the utility at time t for alternative ..E. The choice process M={Mt,t.(0,.)}, is studied, where Mt is the set of utility maximizing alternatives at time t, that is, Mt is the set of ..E at which the sample paths of Yt on E achieve their maximum. Independence properties of Y and M are developed, and general conditions for M to have the Markov property are described. An example of such conditions is that Y have max-stable marginals.