ORIGINAL ARTICLE
On utility maximization under model uncertainty in discrete-time markets
Miklós Rásonyi,
Andrea Meireles-Rodrigues,
Corresponding Author
Miklós Rásonyi
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Correspondence
Miklós Rásonyi, Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, 1053 Budapest, Hungary.
Email: rasonyi@renyi.hu
Search for more papers by this authorAndrea Meireles-Rodrigues
Department of Mathematics, University of York, Heslington, York, United Kingdom
Search for more papers by this authorMiklós Rásonyi,
Andrea Meireles-Rodrigues,
Corresponding Author
Miklós Rásonyi
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Correspondence
Miklós Rásonyi, Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, 1053 Budapest, Hungary.
Email: rasonyi@renyi.hu
Search for more papers by this authorAndrea Meireles-Rodrigues
Department of Mathematics, University of York, Heslington, York, United Kingdom
Search for more papers by this authorAbstract
We study the problem of maximizing terminal utility for an agent facing model uncertainty, in a frictionless discrete-time market with one safe asset and finitely many risky assets. We show that an optimal investment strategy exists if the utility function, defined either on the positive real line or on the whole real line, is bounded from above. We further find that the boundedness assumption can be dropped, provided that we impose suitable integrability conditions, related to some strengthened form of no-arbitrage. These results are obtained in an alternative framework for model uncertainty, where all possible dynamics of the stock prices are represented by a collection of stochastic processes on the same filtered probability space, rather than by a family of probability measures.
REFERENCES
- Acciaio, B., Beiglböck, M., Penkner, F., & Schachermayer, W. (2016). A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Mathematical Finance, 26(2), 233–251.
- Bartl, D. (2019). Exponential utility maximization under model uncertainty for unbounded endowments. Annals of Applied Probability, 29(1), 577–612.
- Bartl, D., Cheridito, P., & Kupper, M. (2019). Robust expected utility maximization with medial limits. Journal of Mathematical Analysis and Applications, 471(1–2), 752–775.
- Beiglböck, M., Henry-Labordère, P., & Penkner, F. (2013). Model-independent bounds for option prices: A mass transport approach. Finance and Stochastics, 17(3), 477–501.
- Blanchard, R., & Carassus, L. (2018). Multiple-priors optimal investment in discrete time for unbounded utility function. Annals of Applied Probability, 28(3), 1856–1892.
- Blanchard, R., & Carassus, L. (2019, October). No-arbitrage with multiple-priors in discrete time. Preprint (version 2). Retrieved from https://arxiv.org/abs/1904.08780.
- Bouchard, B., & Nutz, M. (2015). Arbitrage and duality in nondominated discrete-time models. Annals of Applied Probability, 25(2), 823–859.
- Burzoni, M., Fritelli, M., Hou, Z. Maggis, M., & Oblój, J. (2019). Pointwise arbitrage pricing theory in discrete time. Mathematics of Operations Research, 44(3), 1034–1057.
- Burzoni, M., Fritelli, M., & Maggis, M. (2016). Universal arbitrage aggregator in discrete-time markets under uncertainty. Finance and Stochastics, 20(1), 1–50.
- Carassus, L., & Rásonyi, M. (2007). Optimal strategies and utility-based prices converge when agents' preferences do. Mathematics of Operations Research, 32(1), 102–117.
- Carassus, L., & Rásonyi, M. (2016). Maximization of nonconcave utility functions in discrete-time financial market models. Mathematics of Operations Research, 41(1), 146–173.
- Chau, H. N., & Rásonyi, M. (2019). Robust utility maximization in markets with transaction costs. Finance and Stochastics, 23(3), 677–696.
- Cox, J. C., & Huang, C. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. Journal of Economic Theory, 49(1), 33–83.
- Cvitanić, J., & Karatzas, I. (1992). Convex duality in constrained portfolio optimization. Annals of Applied Probability, 2(4), 767–818.
10.1214/aoap/1177005576 Google Scholar
- Dalang, R. C., Morton, A., & Willinger, W. (1990). Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stochastics and Stochastics Reports, 29(2), 185–201.
10.1080/17442509008833613 Google Scholar
- Davis, M. H. A., & Hobson, D. G. (2007). The range of traded option prices. Mathematical Finance, 17(1), 1–14.
- Dolinsky, Y., & Soner, H. M. (2013). Martingale optimal transport and robust hedging in continuous time. Probability Theory and Related Fields, 160(1–2), 391–427.
- Epstein, L. G., & Ji, S. (2014). Ambiguous volatility, possibility and utility in continuous time. Journal of Mathematical Economics, 50, 269–282.
- Föllmer, H., Schied, A., & Weber, S. (2009). Robust preferences and robust portfolio choice. In P. G. Ciarlet, A. Bensoussan, & Q. Zhang (Eds.), Mathematical modeling and numerical methods in finance (pp. 29–89). Amsterdam: Elsevier.
- Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with non-unique prior. Journal of Mathematical Economics, 18(2), 141–153.
- Imkeller, P., & Perkowski, N. (2015). The existence of dominating local martingale measures. Finance and Stochastics, 19(4), 685–717.
- Jacod, J., & Shiryaev, A. (1998). Local martingales and the fundamental asset pricing theorems in the discrete-time case. Finance and Stochastics, 2(3), 259–273.
10.1007/s007800050040 Google Scholar
- Karatzas, I., Lehoczky, J. P. , & Shreve, S. E. (1987). Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM Journal on Control and Optimization, 25(6), 1557–1586.
- Karatzas, I., Lehoczky, J. P., Shreve, S. E., & Xu, G.-L. (1991). Martingale and duality methods for utility maximization in an incomplete market. SIAM Journal on Control and Optimization, 29(3), 702–730.
- Kramkov, D., & Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Annals of Applied Probability, 9(3), 904–950.
- Merton, R. (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3(4), 373–413.
- Neufeld, A., & Šikić, M. (2018). Robust utility maximization in discrete-time markets with friction. SIAM Journal on Control and Optimization, 56(3), 1912–1937.
- Nutz, M. (2016). Utility maximization under model uncertainty in discrete time. Mathematical Finance, 26(2), 252–268.
- Pliska, S. R. (1986). A stochastic calculus model of continuous trading: Optimal portfolios. Mathematics of Operations Research, 11(2), 371–382.
- Rásonyi, M. (2016). On optimal strategies for utility maximizers in the arbitrage pricing model. International Journal of Theoretical and Applied Finance, 19(7), paper 1650047, 12.
- Rásonyi, M. (2017). Maximizing expected utility in the arbitrage pricing model. Journal of Mathematical Analysis and Applications, 454(1), 127–143.
- Rásonyi, M., & Rodrigues, A. M. (2013). Optimal portfolio choice for a behavioural investor in continuous-time markets. Annals of Finance, 9(2), 291–318.
10.1007/s10436-012-0211-4 Google Scholar
- Rásonyi, M., & Rodríguez-Villarreal, J. G. (2015). Optimal investment under behavioural criteria – A dual approach. Banach Center Publications 104, 167–180.
10.4064/bc104-0-9 Google Scholar
- Rásonyi, M., & Stettner, Ł. (2005). On utility maximization in discrete-time financial market models. Annals of Applied Probability, 15(2), 1367–1395.
- Rásonyi, M., & Stettner, Ł. (2006). On the existence of optimal portfolios for the utility maximization problem in discrete time financial market models. In Y. Kabanov, R. Liptser, & J. Stoyanov (Eds.), From stochastic calculus to mathematical finance (pp. 589–608). Berlin, Heidelberg: Springer.
10.1007/978-3-540-30788-4_29 Google Scholar
- Riedel, F. (2009). Optimal stopping with multiple priors. Econometrica, 77(3), 857–908.
- Schachermayer, W. (1992). A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insurance Mathematics and Economics, 11(4), 249–257.
- Schied, A. (2005). Optimal investments for robust utility functionals in complete market models. Mathematics of Operations Research, 30(3), 750–764.
- Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton, NJ; Woodstock: Princeton University Press.
- Zariphopoulou, T. (1994). Consumption-investment models with constraints. Journal on Control and Optimization, 32(1), 59–85.
