Anna Sulima | Searcharxiv

arXiv2018

Optimal portfolio selection in an Itô-Markov additive market

We study a portfolio selection problem in a continuous-time Itô-Markov additive market with prices of financial assets described by Markov additive processes which combine Lévy processes and regime switching models. Thus the model takes into account two sources of risk: the jump diffusion risk and the regime switching risk. For this reason the market is incomplete. We complete the market by enlarging it with the use of a set of Markovian jump securities, Markovian power-jump securities and impulse regime switching securities. Moreover, we give conditions under which the market is asymptotic-arbitrage-free. We solve the portfolio selection problem in the Itô-Markov additive market for the power utility and the logarithmic utility.

q-fin.PM↗

arXiv2017

A note on chaotic and predictable representations for Itô-Markov additive processes

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.

math.PR↗