Outperforming the market portfolio with a given probability

TL;DR

This paper characterizes the minimal initial capital needed for an investor to outperform the market portfolio with a specified probability, using a nonlinear PDE approach under relaxed market assumptions.

Contribution

It introduces a novel PDE-based framework to determine the minimal capital for outperforming the market with given probability, without requiring an equivalent martingale measure.

Findings

The value function is the smallest nonnegative viscosity supersolution of a nonlinear PDE.

The approach works under the existence of a local martingale deflator, not necessarily an equivalent measure.

Provides a new method to assess outperforming the market probabilistically.

Abstract

Our goal is to resolve a problem proposed by Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.]: to characterize the minimum amount of initial capital with which an investor can beat the market portfolio with a certain probability, as a function of the market configuration and time to maturity. We show that this value function is the smallest nonnegative viscosity supersolution of a nonlinear PDE. As in Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.], we do not assume the existence of an equivalent local martingale measure, but merely the existence of a local martingale deflator.