The Annals of Applied Probability

Optimal arbitrage under model uncertainty

Daniel Fernholz and Ioannis Karatzas

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 23 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1322057320

Digital Object Identifier
doi:10.1214/10-AAP755

Mathematical Reviews number (MathSciNet)
MR2895414

Zentralblatt MATH identifier
1239.60057

Subjects
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]

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

Citation

Fernholz, Daniel; Karatzas, Ioannis. Optimal arbitrage under model uncertainty. Ann. Appl. Probab. 21 (2011), no. 6, 2191--2225. doi:10.1214/10-AAP755. https://projecteuclid.org/euclid.aoap/1322057320


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