Abstract
We study a portfolio selection problem in a continuous-time Itô–Markov additive market with prices of financial assets described by Markov additive processes that 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.
Highlights
The portfolio selection problem is an important issue in financial mathematics
Markowitz (1952) for the first time used the quantitative methods for the optimal portfolio selection problem and proposed the mean-variance approach for portfolio optimization
Explicit solutions for the portfolio selection problem in continuous time were first given by Merton (1971, 1980)
Summary
The portfolio selection problem is an important issue in financial mathematics. The problem is to invest an initial wealth in financial assets so as to maximize the expected utility of the terminal wealth. We consider a market with the prices of financial assets described by Itô–Markov additive processes, which combine Lévy processes and regime switching models. Such a process evolves as an Itô–Lévy process between changes of states of a Markov chain, that is, its parameters depend on the current state of the Markov chain. We show how to complete the Itô–Markov additive market model by adding Markovian jump securities, Markovian power-jump securities and impulse regime switching securities Using these securities, all contingent claims can be replicated by a self-financing portfolio.
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