Outperforming the Market Portfolio with a Given Probability
The Annals of Applied Probability (2012), Vol. 22, No. 4, 1465-1494
30 Pages Posted: 1 Feb 2014
Date Written: June 1, 2011
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
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