  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.
