🤖 AI Summary
This paper extends the Merton investment-consumption problem by imposing a strict benchmark-tracking constraint on wealth: virtual capital injections are permitted to ensure that compensated wealth never falls below a prescribed benchmark, with the objective of maximizing expected utility of consumption net of injection costs. The injection cost equals the expected maximum shortfall of wealth relative to the benchmark. Methodologically, we construct a zero-reflection auxiliary state process, transforming the original problem into a stochastic control problem with upper and lower reflective boundaries. We develop, for the first time, a convex duality theory tailored to state-dependent reflection constraints, yielding explicit feedback-form optimal policies under CRRA utility. For a geometric Brownian motion benchmark, we derive closed-form solutions, quantifying how capital injection incentives affect risky asset allocation and tracking performance, and establishing the fundamental equivalence between injection cost and the expected maximum wealth shortfall.
📝 Abstract
This paper studies a Merton's optimal portfolio and consumption problem in an extended formulation by incorporating the benchmark tracking on the wealth process. We consider a tracking formulation such that the wealth process compensated by a fictitious capital injection outperforms the benchmark at all times. The fund manager aims to maximize the expected utility of consumption deducted by the cost of the capital injection, where the latter term can also be interpreted as the expected largest shortfall of the wealth with reference to the benchmark. By considering an auxiliary state process, we formulate an equivalent stochastic control problem with state reflections at zero. For general utility functions and It^o diffusion benchmark process, we develop a convex duality theorem, new to the literature, to 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.
