The Annals of Applied Probability

Outperforming the market portfolio with a given probability

Erhan Bayraktar, Yu-Jui Huang, and Qingshuo Song

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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.

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Ann. Appl. Probab., Volume 22, Number 4 (2012), 1465-1494.

First available in Project Euclid: 10 August 2012

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Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60H10: Stochastic ordinary differential equations [See also 34F05] 91G99: None of the above, but in this section
Secondary: 60G44: Martingales with continuous parameter 35A02: Uniqueness problems: global uniqueness, local uniqueness, non- uniqueness 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Strict local martingale deflators optimal arbitrage quantile hedging viscosity solutions nonuniqueness of solutions of nonlinear PDEs


Bayraktar, Erhan; Huang, Yu-Jui; Song, Qingshuo. Outperforming the market portfolio with a given probability. Ann. Appl. Probab. 22 (2012), no. 4, 1465--1494. doi:10.1214/11-AAP799.

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