Martingale Approach to Optimal Portfolio-Consumption Problems in Markov-Modulated Pure-Jump Models

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.

Mathematics Subject Classifications:

• Primary 91G10
• Secondary 60J75
• 93E20

Keywords:

• Jump-telegraph model
• Markov-modulated
• Martingale method
• Optimal investment-consumption
• Pure jump model
• Regime switching
• Utility maximization