Chaotic and Predictable Representations for Markov Additive Processes with Levy Modulator

TL;DR

This paper develops martingale and chaotic representations for Markov additive processes with Levy modulator, enabling a deeper understanding of their structure and stochastic calculus applications.

Contribution

It introduces a novel chaotic representation for these processes by orthogonalizing Teugels martingales, extending stochastic calculus tools.

Findings

Martingale representations for Markov additive processes with Levy modulator.

Chaotic representation of square-integrable variables as stochastic integrals.

Predictable representation of square-integrable martingales derived.

Abstract

Our main result is the martingale representations for Markov additive processes where the modulator is a Levy process. These processes have three parts: the modulator, the jumps of the ordinate triggered by the modulator, and the semimartingale part of the ordinate with parameters depending on the modulator. We orthogonalize Teugels martingales constructed from these parts to give a chaotic representation of square-integrable random variables as a sum of stochastic integrals with respect to the orthogonal sequence obtained. Consequently, a predictable representation of square-integrable martingales is derived in terms of the ordinate and the Teugels martingales.