Portfolio Exponential Utility Maximization with Jump Signals

TL;DR

This paper develops a framework for maximizing exponential utility in portfolio management with assets driven by Brownian motion and Poisson jumps, incorporating jump signals into strategies and solving related BSDEs.

Contribution

It introduces a novel BSDE with jumps for utility maximization, including jump signals in strategies, and proves existence of solutions with numerical utility gain analysis.

Findings

Derived an original BSDE with jumps for utility maximization.

Proved existence of solutions to the BSDE.

Numerical experiments show utility gains from jump signals.

Abstract

In this paper, we study the portfolio utility maximization in the case where the risky asset is driven by a Brownian motion and an independent homogeneous Poisson measure, with strategies that may include jump signals. This means that the allowed strategies are no longer predictable but also include the information given by a process driven by the Poisson measure. Using the results of Bank and K{\"o}rber [1], we first express the considered portfolio as semi-martingale processes. We then present the martingale optimality principle for the exponential utility maximization. This allows to derive an original BSDE with jumps and to express the optimal value and an optimal strategy using the solution to this original BSDE. We then prove existence of a solution to the considered BSDE. We finally present some numerical experiments to quantify the gain of utility given by the information from the jump signals.