## The Annals of Applied Probability

### Robust maximization of asymptotic growth under covariance uncertainty

#### Abstract

This paper resolves a question proposed in Kardaras and Robertson [Ann. Appl. Probab. 22 (2012) 1576–1610]: how to invest in a robust growth-optimal way in a market where precise knowledge of the covariance structure of the underlying assets is unavailable. Among an appropriate class of admissible covariance structures, we characterize the optimal trading strategy in terms of a generalized version of the principal eigenvalue of a fully nonlinear elliptic operator and its associated eigenfunction, by slightly restricting the collection of nondominated probability measures.

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 5 (2013), 1817-1840.

Dates
First available in Project Euclid: 28 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1377696299

Digital Object Identifier
doi:10.1214/12-AAP887

Mathematical Reviews number (MathSciNet)
MR3114918

Zentralblatt MATH identifier
1287.60081

#### Citation

Bayraktar, Erhan; Huang, Yu-Jui. Robust maximization of asymptotic growth under covariance uncertainty. Ann. Appl. Probab. 23 (2013), no. 5, 1817--1840. doi:10.1214/12-AAP887. https://projecteuclid.org/euclid.aoap/1377696299

#### References

• [1] Arapostathis, A., Borkar, V. S. and Ghosh, M. K. (2012). Ergodic Control of Diffusion Processes. Encyclopedia of Mathematics and Its Applications 143. Cambridge Univ. Press, Cambridge.
• [2] Avellaneda, M., Levy, A. and Paras, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2 73–88.
• [3] Bayraktar, E. and Huang, Y. J. (2011). On the multi-dimensional controller and stopper games. Technical report, Univ. Michigan. Available at http://arxiv.org/abs/1009.0932.
• [4] Birindelli, I. and Demengel, F. (2007). Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Commun. Pure Appl. Anal. 6 335–366.
• [5] Birindelli, I. and Demengel, F. (2010). Eigenfunctions for singular fully nonlinear equations in unbounded domains. NoDEA Nonlinear Differential Equations Appl. 17 697–714.
• [6] Borkar, V. S. (2006). Ergodic control of diffusion processes. In International Congress of Mathematicians. Vol. III 1299–1309. Eur. Math. Soc., Zürich.
• [7] Brown, A. L. (1989). Set valued mappings, continuous selections, and metric projections. J. Approx. Theory 57 48–68.
• [8] Busca, J., Esteban, M. J. and Quaas, A. (2005). Nonlinear eigenvalues and bifurcation problems for Pucci’s operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 187–206.
• [9] Caffarelli, L. A. and Cabré, X. (1995). Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications 43. Amer. Math. Soc., Providence, RI.
• [10] Denis, L., Hu, M. and Peng, S. (2011). Function spaces and capacity related to a sublinear expectation: Application to $G$-Brownian motion paths. Potential Anal. 34 139–161.
• [11] Denis, L. and Martini, C. (2006). A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16 827–852.
• [12] Fernholz, D. and Karatzas, I. (2011). Optimal arbitrage under model uncertainty. Ann. Appl. Probab. 21 2191–2225.
• [13] Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd ed. Stochastic Modelling and Applied Probability 25. Springer, New York.
• [14] Fujisaki, M. (1999). On probabilistic approach to the eigenvalue problem for maximal elliptic operator. Osaka J. Math. 36 981–992.
• [15] Gilbarg, D. and Trudinger, N. S. (2001). Elliptic Partial Differential Equations of Second Order. Springer, Berlin. Reprint of the 1998 edition.
• [16] Kardaras, C. and Robertson, S. (2012). Robust maximization of asymptotic growth. Ann. Appl. Probab. 22 1576–1610.
• [17] Kawohl, B. and Kutev, N. (2007). Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations. Comm. Partial Differential Equations 32 1209–1224.
• [18] Kelley, J. L. (1955). General Topology. Van Nostrand, New York.
• [19] Lyons, T. J. (1995). Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2 117–133.
• [20] McCoy, J. W. (1965). An extension of the concept of $L_{n}$ sets. Proc. Amer. Math. Soc. 16 177–180.
• [21] Meyer, C. (2000). Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia, PA.
• [22] Michael, E. (1956). Continuous selections. I. Ann. of Math. (2) 63 361–382.
• [23] Nutz, M. (2012). A quasi-sure approach to the control of non-Markovian stochastic differential equations. Electron. J. Probab. 17 23.
• [24] Nutz, M. and Soner, H. M. (2010). Superhedging and dynamic risk measures under volatility uncertainty. Technical report, ETH Zürich. Available at http://arxiv.org/abs/1011.2958.
• [25] Peng, S. (2007). $G$-Brownian motion and dynamic risk measure under volatility uncertainty. Technical report, Shandong Univ. Available at http://arxiv.org/abs/0711.2834.
• [26] Pinsky, R. G. (1995). Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics 45. Cambridge Univ. Press, Cambridge.
• [27] Pucci, C. (1966). Maximum and minimum first eigenvalues for a class of elliptic operators. Proc. Amer. Math. Soc. 17 788–795.
• [28] Pucci, C. (1966). Operatori ellittici estremanti. Ann. Mat. Pura Appl. (4) 72 141–170.
• [29] Quaas, A. and Sirakov, B. (2008). Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators. Adv. Math. 218 105–135.
• [30] Safonov, M. V. (1988). Classical solution of second-order nonlinear elliptic equations. Izv. Akad. Nauk SSSR Ser. Mat. 52 1272–1287, 1328.
• [31] Soner, H. M., Touzi, N. and Zhang, J. (2011). Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16 1844–1879.