A large deviations approach to optimal long term investment

We consider an investment model where the objective is to overperform a given benchmark or index. We study this portfolio management problem for a long term horizon. This asymptotic criterion leads to a large deviation probability control problem. Its dual problem is an ergodic risk sensitive control problem on the optimal logarithmic moment generating function that is explicitly derived. A careful study of its domain and its behavior at the boundary of the domain is required. We then use large deviations techniques for stating the value function of this criterion of outperformance management. This provides in turn an objective probabilistic interpretation of the usually subjective degree of risk aversion in CRRA utility function.