Maximizing The Growth Rate Under Risk Constraints

We investigate the ergodic problem of growth‐rate maximization under a class of risk constraints in the context of incomplete, Itô‐process models of financial markets with random ergodic coefficients. Including value‐at‐risk, tail‐value‐at‐risk, and limited expected loss, these constraints can be both wealth‐dependent (relative) and wealth‐independent (absolute). The optimal policy is shown to exist in an appropriate admissibility class, and can be obtained explicitly by uniform, state‐dependent scaling down of the unconstrained (Merton) optimal portfolio. This implies that the risk‐constrained wealth‐growth optimizer locally behaves like a constant relative risk aversion (CRRA) investor, with the relative risk‐aversion coefficient depending on the current values of the market coefficients.