Open Access
December 2011 Optimal arbitrage under model uncertainty
Daniel Fernholz, Ioannis Karatzas
Ann. Appl. Probab. 21(6): 2191-2225 (December 2011). DOI: 10.1214/10-AAP755


In an equity market model with “Knightian” uncertainty regarding the relative risk and covariance structure of its assets, we characterize in several ways the highest return relative to the market that can be achieved using nonanticipative investment rules over a given time horizon, and under any admissible configuration of model parameters that might materialize. One characterization is in terms of the smallest positive supersolution to a fully nonlinear parabolic partial differential equation of the Hamilton–Jacobi–Bellman type. Under appropriate conditions, this smallest supersolution is the value function of an associated stochastic control problem, namely, the maximal probability with which an auxiliary multidimensional diffusion process, controlled in a manner which affects both its drift and covariance structures, stays in the interior of the positive orthant through the end of the time-horizon. This value function is also characterized in terms of a stochastic game, and can be used to generate an investment rule that realizes such best possible outperformance of the market.


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Daniel Fernholz. Ioannis Karatzas. "Optimal arbitrage under model uncertainty." Ann. Appl. Probab. 21 (6) 2191 - 2225, December 2011.


Published: December 2011
First available in Project Euclid: 23 November 2011

zbMATH: 1239.60057
MathSciNet: MR2895414
Digital Object Identifier: 10.1214/10-AAP755

Primary: 60H10 , 91B28
Secondary: 35B50 , 60G44 , 60J70

Keywords: Arbitrage , fully nonlinear parabolic equations , maximal containment probability , minimal solutions , model uncertainty , Robust portfolio choice , Stochastic control , stochastic game

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.21 • No. 6 • December 2011
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