Utility maximization under model uncertainty in discrete time. (English) Zbl 1378.91114
This paper establishes the existence of an optimal portfolio in a general discrete-time model with uncertainty about the underlying probabilistic model. A key feature is that the scenario set of possible probability measures need not be dominated by a single reference model. Existence of the optimizer is shown under a notion of no-arbitrage introduced in [B. Bouchard and M. Nutz, Ann. Appl. Probab. 25, No. 2, 823–859 (2015; Zbl 1322.60045)] for a utility function bounded from above. A counterexample illustrates that additional integrability assumptions are needed to obtain existence in the unbounded case.
Reviewer: Johannes Muhle-Karbe (Pittsburgh)
MSC:
| 91G10 | Portfolio theory |
| 60G44 | Martingales with continuous parameter |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
Citations:
Zbl 1322.60045References:
| [1] | Acciaio, B., M.Beiglböck, F.Penkner, and W.Schachermayer (2016): A Model‐Free Version of the Fundamental Theorem of Asset Pricing and the Super‐Replication Theorem. Math. Finance26(2), 233-251. · Zbl 1378.91129 |
| [2] | Aliprantis, C. D., and K. C.Border (2006): Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed., Berlin: Springer. · Zbl 1156.46001 |
| [3] | Bayraktar, E., and Y.‐J.Huang (2013): Robust Maximization of Asymptotic Growth under Covariance Uncertainty, Ann. Appl. Probab.23(5), 1721-2160. |
| [4] | Beiglböck, M., P.Henry‐Labordère, and F.Penkner (2013): Model‐Independent Bounds for Option Prices: A Mass Transport Approach, Finance Stoch.17(3), 477-501. · Zbl 1277.91162 |
| [5] | Bertsekas, D. P., and S. E.Shreve (1978): Stochastic Optimal Control. The Discrete‐Time Case. New York: Academic Press. · Zbl 0471.93002 |
| [6] | Bouchard, B., and M.Nutz (2015): Arbitrage and Duality in Nondominated Discrete‐Time Models. Ann. Appl. Probab25(2), 823-859. · Zbl 1322.60045 |
| [7] | Cont, R. (2006): Model Uncertainty and Its Impact on the Pricing of Derivative Instruments, Math. Finance16(3), 519-547. · Zbl 1133.91413 |
| [8] | Cox, A. M. G., and J.Obłój (2011): Robust Pricing and Hedging of Double No‐Touch Options, Finance Stoch.15(3), 573-605. · Zbl 1303.91171 |
| [9] | Delbaen, F., and W.Schachermayer (1994): A General Version of the Fundamental Theorem of Asset Pricing. Math. Ann.300, 463-520. · Zbl 0865.90014 |
| [10] | Denis, L., and M.Kervarec (2013): Optimal Investment under Model Uncertainty in Nondominated Models. SIAM J. Control Optim.51(3), 1803-1822. · Zbl 1331.60076 |
| [11] | Denis, L., and C.Martini (2006): A Theoretical Framework for the Pricing of Contingent Claims in the Presence of Model Uncertainty. Ann. Appl. Probab.16(2), 827-852. · Zbl 1142.91034 |
| [12] | Dolinsky, Y., M.Nutz, and H. M.Soner (2012): Weak Approximation of G‐Expectations. Stoch. Process. Appl.122(2), 664-675. · Zbl 1259.60073 |
| [13] | Dolinsky, Y., and H. M.Soner (2014): Martingale Optimal Transport and Robust Hedging in Continuous Time. Probab. Theory Related Fields160(1), 391-427. · Zbl 1305.91215 |
| [14] | Donoghue, W. F. (1969): Distributions and Fourier Transforms, Vol. 32: Pure and Applied Mathematics. New York: Academic Press. · Zbl 0188.18102 |
| [15] | Fernholz, D., and I.Karatzas (2011): Optimal Arbitrage under Model Uncertainty. Ann. Appl. Probab.21(6), 2191-2225. · Zbl 1239.60057 |
| [16] | Föllmer, H., and A.Schied (2004): Stochastic Finance: An Introduction in Discrete Time, 2nd ed., Berlin: W. de Gruyter. · Zbl 1126.91028 |
| [17] | Galichon, A., P.Henry‐Labordère, and N.Touzi (2014): A Stochastic Control Approach to No‐Arbitrage Bounds Given Marginals, with an Application to Lookback Options. Ann. Appl. Probab.24(1), 312-336. · Zbl 1285.49012 |
| [18] | Gilboa, I., and D.Schmeidler (1989): Maxmin Expected Utility with Non‐Unique Prior. J. Math. Econ.18(2), 141-153. · Zbl 0675.90012 |
| [19] | Karatzas, I., and S. E.Shreve (1998): Methods of Mathematical Finance. New York: Springer. · Zbl 0941.91032 |
| [20] | Kramkov, D., and W.Schachermayer (1999): The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets. Ann. Appl. Probab.9(3), 904-950. · Zbl 0967.91017 |
| [21] | Leese, S. J. (1978): Measurable Selections and the Uniformization of Souslin Sets. Amer. J. Math.100(1), 19-41. · Zbl 0384.28005 |
| [22] | Matoussi, A., D.Possamai, and C.Zhou (2015): Robust Utility Maximization in Non‐Dominated Models with 2BSDEs: The Uncertain Volatility Model. Math. Finance25(2), 258-287. · Zbl 1335.91067 |
| [23] | Neufeld, A., and M.Nutz (2013): Superreplication under Volatility Uncertainty for Measurable Claims. Electron. J. Probab.18(48), 1-14. · Zbl 1282.91360 |
| [24] | Nutz, M. (2013): Random G‐Expectations. Ann. Appl. Probab.23(5), 1755-1777. · Zbl 1273.93178 |
| [25] | Nutz, M., and R.van Handel (2013): Constructing Sublinear Expectations on Path Space. Stochastic Process. Appl.123(8), 3100-3121. · Zbl 1285.93104 |
| [26] | Peng, S. (2010): Nonlinear Expectations and Stochastic Calculus under Uncertainty. Preprint arXiv:1002.4546v1. |
| [27] | Quenez, M.‐C. (2004): Optimal Portfolio in a Multiple‐Priors Model, in Seminar on Stochastic Analysis, Random Fields and Applications IV, R. Dalang, M. Dozzi, Marco, and F. Russo, eds. Vol. 58: Progress in Probability, Basel: Birkhäuser, pp. 291-321. · Zbl 1061.93100 |
| [28] | Rásonyi, M., and L.Stettner (2005): On Utility Maximization in Discrete‐Time Financial Market Models. Ann. Appl. Probab.15(2), 1367-1395. · Zbl 1137.93423 |
| [29] | Rásonyi, M., and L.Stettner (2006): On the Existence of Optimal Portfolios for the Utility Maximization Problem in Discrete Time Financial Market Models, in From Stochastic Calculus to Mathematical Finance, Y. Kabanov, R. Liptser, and J. Stoyanov, eds. Berlin: Springer, pp. 589-608. · Zbl 1098.93038 |
| [30] | Schäl, M. (2000): Portfolio Optimization and Martingale Measures, Math. Finance10(2), 289-303. · Zbl 1016.91051 |
| [31] | Schied, A. (2006): Risk Measures and Robust Optimization Problems, Stoch. Models22(4), 753-831. · Zbl 1211.91151 |
| [32] | Soner, H. M., N.Touzi, and J.Zhang (2011): Martingale Representation Theorem for the G‐Expectation. Stoch. Process. Appl.121(2), 265-287. · Zbl 1228.60070 |
| [33] | Soner, H. M., N.Touzi, and J.Zhang (2013): Dual Formulation of Second Order Target Problems. Ann. Appl. Probab.23(1), 308-347. · Zbl 1293.60063 |
| [34] | Talay, D., and Z.Zheng (2002): Worst Case Model Risk Management, Finance Stoch.6(4), 517-537. · Zbl 1039.91031 |
| [35] | Tevzadze, R., T.Toronjadze, and T.Uzunashvili (2013): Robust Utility Maximization for a Diffusion Market Model with Misspecified Coefficients, Finance Stoch.17(3), 535-563. · Zbl 1270.91027 |
| [36] | Wald, A. (1950): Statistical Decision Functions. New York: John Wiley & Sons Inc. · Zbl 0040.36402 |
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