Exponential families of stochastic processes and Lévy processes

Abstract

It is shown that a class of Lévy processes (processes with independent stationary increments) is connected in a natural way to many exponential families of continuous-time stochastic processes. Specifically, the canonical process has independent increments if the family has a nonempty kernel. In many other exponential models the canonical process can be obtained from a Lévy process by a stochastic time transformation. The basic observed process need not have independent increments. It can, for instance, be a diffusion-type process or a counting process. We study such properties of exponential families of processes that concern the attached class of Lévy processes or can be derived from it. In particular, likelihood theory and maximum likelihood estimation is considered. A thorough discussion is given of exponential families of Lévy processes. We also consider the construction of exponential families of processes with a more complicated dynamics by stochastic time transformation of exponential families of Lévy processes.