The Annals of Applied Probability

Optimal arbitrage under model uncertainty

Daniel Fernholz and Ioannis Karatzas

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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.

Article information

Ann. Appl. Probab., Volume 21, Number 6 (2011), 2191-2225.

First available in Project Euclid: 23 November 2011

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 91B28
Secondary: 60G44: Martingales with continuous parameter 35B50: Maximum principles 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Robust portfolio choice model uncertainty arbitrage fully nonlinear parabolic equations minimal solutions maximal containment probability stochastic control stochastic game


Fernholz, Daniel; Karatzas, Ioannis. Optimal arbitrage under model uncertainty. Ann. Appl. Probab. 21 (2011), no. 6, 2191--2225. doi:10.1214/10-AAP755.

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