Optimal portfolio selection in an It\^o-Markov additive market

TL;DR

This paper addresses optimal portfolio selection in a complex continuous-time market model that incorporates both jump diffusion and regime switching risks, proposing market completion techniques and solving utility maximization problems.

Contribution

It introduces a novel framework combining Markov additive processes with market completion methods for portfolio optimization.

Findings

Market is incomplete due to jump and regime risks.

Market can be completed with specific securities.

Optimal portfolios are derived for power and logarithmic utilities.

Abstract

We study a portfolio selection problem in a continuous-time It\^o-Markov additive market with prices of financial assets described by Markov additive processes which combine L\'evy 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 It\^o-Markov additive market for the power utility and the logarithmic utility.