Utility maximization in pure-jump models driven by marked point processes and nonlinear wealth dynamics

TL;DR

This paper develops a framework for optimal investment strategies in markets with jumps and nonlinear wealth dynamics, providing conditions for optimal policies and explicit solutions in certain models.

Contribution

It introduces a novel approach combining martingale and convex duality techniques to handle complex jump-driven market models with nonlinear wealth effects.

Findings

Established sufficient conditions for optimal policy existence.

Derived closed-form solutions for pure-jump models with Markov-modulated jump sizes.

Extended utility maximization to models with differential interest rates and negative rebates.

Abstract

We explore martingale and convex duality techniques to study optimal investment strategies that maximize expected risk-averse utility from consumption and terminal wealth. We consider a market model with jumps driven by (multivariate) marked point processes and so-called non-linear wealth dynamics which allows to take account of relaxed assumptions such as differential borrowing and lending interest rates or short positions with cash collateral and negative rebate rates. We give suffcient conditions for existence of optimal policies for agents with logarithmic and CRRA power utility. We find closed-form solutions for the optimal value function in the case of pure-jump models with jump-size distributions modulated by a two-state Markov chain.