An Extended Merton Problem With Relaxed Benchmark Tracking

This paper studies Merton's problem in an extended formulation by incorporating a benchmark tracking on the wealth process. We consider a tracking formulation where the fund manager aims to maximize the trade‐off between the expected utility of consumption and the expected largest shortfall in wealth relative to the benchmark level. Equivalently, the problem can be interpreted as a mixed stochastic control problem if a fictitious capital injection singular control is allowed, subject to the dynamic constraint that the wealth process compensated by the costly capital injection outperforms the benchmark at all times. By considering an auxiliary state process, we formulate an equivalent stochastic control problem with state reflections at zero. For general utility functions and Itô's diffusion benchmark process, we develop a convex duality theorem, new to the literature, for the auxiliary stochastic control problem with state reflections in which the dual process also exhibits reflections from above. For CRRA utility and geometric Brownian motion benchmark process, we further derive the optimal portfolio and consumption in feedback form using the new duality theorem, allowing us to discuss some interesting financial implications induced by the additional risk‐taking from the capital injection and the goal of tracking.