Outperformance Portfolio Optimization via the Equivalence of Pure and Randomized Hypothesis Testing

TL;DR

This paper explores maximizing the probability of outperforming a benchmark through dynamic trading, establishing a connection between pure and randomized hypothesis testing, and providing solutions in both complete and incomplete market settings.

Contribution

It introduces a dual representation linking pure and randomized hypothesis testing in portfolio optimization and offers explicit solutions in certain market models.

Findings

Derived a dual representation for outperformance probability.

Provided a closed-form solution for beating a leveraged ETF in complete markets.

Characterized the problem via HJB PDE in incomplete stochastic factor models.

Abstract

We study the portfolio problem of maximizing the outperformance probability over a random benchmark through dynamic trading with a fixed initial capital. Under a general incomplete market framework, this stochastic control problem can be formulated as a composite pure hypothesis testing problem. We analyze the connection between this pure testing problem and its randomized counterpart, and from latter we derive a dual representation for the maximal outperformance probability. Moreover, in a complete market setting, we provide a closed-form solution to the problem of beating a leveraged exchange traded fund. For a general benchmark under an incomplete stochastic factor model, we provide the Hamilton-Jacobi-Bellman PDE characterization for the maximal outperformance probability.