2016/12/29 by Palmowski, Zbigniew, Stettner, Łukasz, Sulima, Anna · 1 citation
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1612.09216
IIn this paper we provide predictable and chaotic representations for Itô-Markov additive processes X. Such a process is governed by a finite-state CTMC 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.