Outperforming the Market Portfolio with a Given Probability

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.

Keywords: strict local martingale deflators, optimal arbitrage, quantile hedging, viscosity solutions, nonuniqueness of solutions of nonlinear PDEs

Suggested Citation: Suggested Citation

Bayraktar, Erhan and Huang, Yu-Jui and Song, Qingshuo, Outperforming the Market Portfolio with a Given Probability (June 1, 2011). The Annals of Applied Probability (2012), Vol. 22, No. 4, 1465-1494, Available at SSRN: https://ssrn.com/abstract=2388022