This paper studies a Merton's optimal portfolio and consumption problem in an
extended formulation incorporating the tracking of a benchmark process
described by a geometric Brownian motion. We consider a relaxed tracking
formulation such that that the wealth process compensated by a fictitious
capital injection outperforms the external 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 regarded as
the expected largest shortfall with reference to the benchmark. By introducing
an auxiliary state process with reflection, we formulate and tackle an
equivalent stochastic control problem by means of the dual transform and
probabilistic representation, where the dual PDE can be solved explicitly. On
the strength of the closed-form results, we can derive and verify the feedback
optimal control in the semi-analytical form for the primal control problem,
allowing us to observe and discuss some new and interesting financial
implications on portfolio and consumption decision making induced by the
additional risk-taking in capital injection and the goal of tracking.Comment: Keywords: Benchmark tracking, capital injection, expected largest
shortfall, consumption and portfolio choice, Neumann boundary conditio