Martingale approach to optimal portfolio-consumption problems in Markov-modulated pure-jump models

TL;DR

This paper develops a martingale approach to determine optimal investment and consumption strategies in a regime-switching pure-jump market, providing explicit solutions for certain utility functions.

Contribution

It introduces a novel martingale framework for Markov-modulated jump models and derives explicit optimal policies for specific utility functions.

Findings

Closed-form solutions for optimal policies with logarithmic and CRRA utility.

Sufficient conditions for the existence of optimal investment strategies.

Explicit formulas for moments of Markov-modulated telegraph processes.

Abstract

We study optimal investment strategies that maximize expected utility from consumption and terminal wealth in a pure-jump asset price model with Markov-modulated (regime switching) jump-size distributions. We give sufficient conditions for existence of optimal policies and find closed-form expressions for the optimal value function for agents with logarithmic and fractional power (CRRA) utility in the case of two-state Markov chains. The main tools are convex duality techniques, stochastic calculus for pure-jump processes and explicit formulae for the moments of telegraph processes with Markov-modulated random jumps.