A note on chaotic and predictable representations

Zbigniew Palmowski , Łukasz Stettner , Anna Sulima

Abstract

In this article, we provide predictable and chaotic representations for Itô–Markov additive processes X. Such a process is governed by a finitestate continuous time Markov chain J which allows one to modify the parameters of the Itô-jump process (in so-called regime switchingmanner). 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 squareintegrable random variable is given as a sum of stochastic integrals with respect to someexplicitly constructed orthogonalmartingales.We identify the predictable representation of a square-integrablemartingale 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
Author Zbigniew Palmowski - Wrocław University of Science and Technology (PWr)
Zbigniew Palmowski,,
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, Łukasz Stettner - Instytut Matematyczny (IM PAN) [Polish Academy of Sciences (PAN)]
Łukasz Stettner,,
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, Anna Sulima (MISaF / IZM / DoEaOR)
Anna Sulima,,
- Department of Econometrics and Operations Research
Journal seriesStochastic Analysis and Applications, ISSN 0736-2994, e-ISSN 1532-9356, (A 20 pkt)
Issue year2018
Vol36
No4
Pages622-638
ASJC Classification2604 Applied Mathematics; 1804 Statistics, Probability and Uncertainty; 2613 Statistics and Probability
DOIDOI:10.1080/07362994.2018.1434417
URL https://doi.org/10.1080/07362994.2018.1434417
Languageen angielski
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Palmowski_Stettner_Sulima_A_note_ on_ chaotic2018.pdf 686,28 KB
Score (nominal)20
Score sourcejournalList
Publication indicators Scopus Citations = 1; Scopus SNIP (Source Normalised Impact per Paper): 2017 = 0.703; WoS Impact Factor: 2017 = 0.541 (2) - 2017=0.69 (5)
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