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This paper studies a Merton's optimal portfolio and consumption problem in anextended formulation incorporating the tracking of a benchmark processdescribed by a geometric Brownian motion. We consider a relaxed trackingformulation such that that the wealth process compensated by a fictitiouscapital injection outperforms the external benchmark at all times. The fundmanager aims to maximize the expected utility of consumption deducted by thecost of the capital injection, where the latter term can also be regarded asthe expected largest shortfall with reference to the benchmark. By introducingan auxiliary state process with reflection, we formulate and tackle anequivalent stochastic control problem by means of the dual transform andprobabilistic representation, where the dual PDE can be solved explicitly. Onthe strength of the closed-form results, we can derive and verify the feedbackoptimal control in the semi-analytical form for the primal control problem,allowing us to observe and discuss some new and interesting financialimplications on portfolio and consumption decision making induced by theadditional risk-taking in capital injection and the goal of tracking.

An extended Merton problem with relaxed benchmark tracking