Representations and isomorphism identities for infinitely divisible processes

TL;DR

This paper develops isomorphism identities for nonlinear functionals of infinitely divisible processes, extending classical formulas to a broader class of stochastic processes with applications to various examples.

Contribution

It introduces a general framework for isomorphism identities for infinitely divisible processes, including their Lévy measure representations and applications to specific process examples.

Findings

Derived isomorphism identities for nonlinear functionals

Provided Lévy measure representations for general infinitely divisible processes

Illustrated applications on squared Bessel, Feller diffusions, and permanental processes

Abstract

We propose isomorphism type identities for nonlinear functionals of general infinitely divisible processes. Such identities can be viewed as an analogy of the Cameron-Martin formula for Poissonian infinitely divisible processes but with random translations. The applicability of these tools relies on a precise understanding of L\'evy measures of infinitely divisible processes and their representations, which are developed here in full generality. We illustrate this approach on examples of squared Bessel processes, Feller diffusions, permanental processes, as well as L\'evy processes.