A note on chaotic and predictable representations for Itô–Markov additive processes. Issue 4 (4th July 2018)

• Record Type: Journal Article
• Authors: Palmowski, Zbigniew; Stettner, Łukasz; Sulima, Anna
• Is Part Of: Stochastic analysis and applications. Volume 36:Issue 4(2018)
• Page Start: 622
• Page End: 638
• Publication Date: 2018-07-04
• Subjects: Markov additive processes -- martingale representation -- power-jump process -- orthogonal polynomials -- stochastic integral -- Brownian motion -- regime switching -- complete market; 60J30 -- 60H05; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2018.1434417
• Languages: English
• ISSNs: 0736-2994

Abstract

ABSTRACT: In this article, we provide predictable and chaotic representations for Itô–Markov additive processes X . Such a process is governed by a finite-state continuous time Markov chain J which allows one to modify the parameters of the Itô-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 Itô–Lévy 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.