Optimal Investment with High-Watermark Fee in a Multidimensional Jump Diffusion Model

Karel Janecek, Zheng Li, Mihai Sîrbu

Abstract: We study an optimal investment and consumption problem on infinite horizon, under the assumption that one of the investment opportunities is a fund charging high-watermark fees. The fund and the additional risky assets follow a multidimensional geometric Lévy structure. The interest rate is constant and the utility function has constant relative risk aversion. Identifying the wealth of the investor together with the distance to paying fees as the appropriate states, we obtain a two-dimensional stochastic control problem with both jumps and reflection. We derive the Hamilton--Jacobi--Bellman integro-differential equation, reduce it to one dimension, and then show it has a smooth solution. Using verification arguments the optimal strategies are obtained in feedback form. Some numerical results display the impact of the fees on the investor.