A note on chaotic and predictable representations for It\^o-Markov additive processes

TL;DR

This paper develops predictable and chaotic representations for Itô-Markov additive processes, enabling better modeling of regime-switching dynamics and facilitating derivative pricing in incomplete markets.

Contribution

It introduces new chaotic and predictable representations for Itô-Markov additive processes, generalizing previous results and aiding financial modeling.

Findings

Derived chaotic representation as sum of stochastic integrals with orthogonal martingales

Identified predictable representation involving Brownian motion and jump martingales

Facilitated pricing of derivatives in incomplete markets using the new representations

Abstract

IIn this paper we provide predictable and chaotic representations for It\^{o}-Markov additive processes $X$. Such a process is governed by a finite-state CTMC $J$ which allows one to modify the parameters of the It\^{o}-jump process (in so-called regime switching manner). In addition, the transition of $J$ triggers the jump of $X$ distributed depending on the states of $J$ just prior to the transition. This family of processes includes Markov modulated It\^{o}-L\'evy processes and Markov additive processes. The derived chaotic representation of a square-integrable random variable is given as a sum of stochastic integrals with respect to some explicitly constructed orthogonal martingales. We identify the predictable representation of a square-integrable martingale as a sum of stochastic integrals of predictable processes with respect to Brownian motion and power-jumps martingales related to all the jumps appearing in the model. This result generalizes the seminal result of Jacod-Yor and is of importance in financial mathematics. The derived representation then allows one to enlarge the incomplete market by a series of power-jump assets and to price all market-derivatives.