Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T17:28:17.320Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal consumption with Hindy–Huang–Kreps preferences under nonlinear expectations

Part of: Mathematical economics Stochastic analysis Stochastic systems and control Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 June 2022

Giorgio Ferrari ,
Hanwu Li Open the ORCID record for Hanwu Li [Opens in a new window]  and
Frank Riedel
Giorgio Ferrari*
Affiliation:
Bielefeld University
Hanwu Li*
Affiliation:
Shandong University
Frank Riedel*
Affiliation:
Bielefeld University and University of Johannesburg
*
*Postal address: Center for Mathematical Economics, Universtätstrasse 25, Bielefeld, Germany.
***Postal address: Research Center for Mathematics and Interdisciplinary Sciences, Binhai Rd 72, Qingdao, China. Email address: lihanwu11@163.com
*Postal address: Center for Mathematical Economics, Universtätstrasse 25, Bielefeld, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study an intertemporal consumption and portfolio choice problem under Knightian uncertainty in which agent’s preferences exhibit local intertemporal substitution. We also allow for market frictions in the sense that the pricing functional is nonlinear. We prove existence and uniqueness of the optimal consumption plan, and we derive a set of sufficient first-order conditions for optimality. With the help of a backward equation, we are able to determine the structure of optimal consumption plans. We obtain explicit solutions in a stationary setting in which the financial market has different risk premia for short and long positions.

Keywords

Knightian uncertaintyg-expectationambiguity aversionsingular stochastic control

MSC classification

Primary: 93E20: Optimal stochastic control 91B42: Consumer behavior, demand theory
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 65C30: Stochastic differential and integral equations
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 4 , December 2022 , pp. 1222 - 1251
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alvarez, O. (1994). A singular stochastic control problem in an unbounded domain. Commun. Partial Differential Equat. 19, 2075–2089.CrossRefGoogle Scholar
Bank, P. and El Karoui, N. (2004). A stochastic representation theorem with applications to optimization and obstacle problems. Ann. Prob. 32, 1030–1067.CrossRefGoogle Scholar
Bank, P. and Kauppila, H. (2017). Convex duality for stochastic singular control problems. Ann. Appl. Prob. 27, 485–516.CrossRefGoogle Scholar
Bank, P. and Riedel, F. (2000). Non-time additive utility optimization—the case of certainty. J. Math. Econom. 33, 271–290.CrossRefGoogle Scholar
Bank, P. and Riedel, F. (2001). Optimal consumption choice with intertemporal substitution. Ann. Appl. Prob. 11, 750–788.CrossRefGoogle Scholar
Beissner, P. and Riedel, F. (2019). Equilibria under Knightian price uncertainty. Econometrica 87, 37–64.CrossRefGoogle Scholar
Benth, F. E., Karlsen, K. H. and Reikvam, K. (2001). Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution. Fin. Stoch. 5, 447–467.CrossRefGoogle Scholar
Benth, F. E., Karlsen, K. H. and Reikvam, K. (2001). Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: a viscosity solution approach. Finance Stoch. 5, 275–303.CrossRefGoogle Scholar
Benth, F. E., Karlsen, K. H. and Reikvam, K. (2001). Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs. Stochastics 74, 517–569.Google Scholar
Chen, Z. and Epstein, L. (2002). Ambiguity, risk and asset returns in continuous time. Econometrica 70, 1403–1443.CrossRefGoogle Scholar
Chung, K. L. (2001). A Course in Probability Theory. Academic Press, San Diego.Google Scholar
Coquet, F., Hu, Y., Mémin, J. and Peng, S. (2002). Filtration-consistent nonlinear expectations and related g-expectations. Prob. Theory Relat. Fields 123, 1–27.CrossRefGoogle Scholar
Cvitanić, J. and Karatzas, I. (1993). Hedging contingent claims with constrained portfolios. Ann. Appl. Prob. 3, 652–681.CrossRefGoogle Scholar
El Karoui, N., Peng, S. and Quenez, M. (1997). Backward stochastic differential equations in finance. Math. Finance 7, 1–71.CrossRefGoogle Scholar
Englezos, N. and Karatzas, I. (2009). Utility maximization with habit formation: dynamic programming and stochastic PDEs. SIAM J. Control Optimization 48, 481–520.CrossRefGoogle Scholar
Ferrari, G. (2015). On an integral equation for the free-boundary of stochastic, irreversible investment problems. Ann. Appl. Prob. 25, 150–176.CrossRefGoogle Scholar
Ferrari, G., Li, H. and Riedel, F. (2022). A Knightian irreversible investment problem. To appear in J. Math. Anal. Appl. 507.CrossRefGoogle Scholar
Hindy, A. and Huang, C. F. (1992). Intertemporal preferences for uncertain consumption: a continuous time approach. Econometrica 60, 781–801.CrossRefGoogle Scholar
Hindy, A. and Huang, C. F. (1993). Optimal consumption and portfolio rules with durability and local substitution. Econometrica 61, 85–122.CrossRefGoogle Scholar
Hindy, A., Huang, C. F. and Kreps, D. (1992). On intertemporal preference in continuous time: the case of certainty. J. Math. Econom. 21, 401–440.CrossRefGoogle Scholar
Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Springer, Berlin.CrossRefGoogle Scholar
Jouini, E. and Kallal, H. (1995). Arbitrage in securities markets with short-sales constraints. Math. Finance 5, 197–232.CrossRefGoogle Scholar
Kabanov, Y. (1999). Hedging and liquidation under transaction costs in currency markets. Finance Stoch. 2, 237–248.CrossRefGoogle Scholar
Komlós, J. (1967). A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18, 217–229.CrossRefGoogle Scholar
Maccheroni, F., Marinacci, M. and Rustichini, A. (2006) Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74, 1447–1498.CrossRefGoogle Scholar
Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: the continuous case. Rev. Econom. Statist. 51, 247–257.CrossRefGoogle Scholar
Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3, 373–413.CrossRefGoogle Scholar
Pardoux, E. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14, 55–61.CrossRefGoogle Scholar
Peng, S. (1997). BSDE and related g-expectations. In Backward Stochastic Differential Equations, ed. N. El Karoui and L. Mazliak, Longman, Harlow, pp. 141–159.Google Scholar
Riedel, F. (2009). Optimal consumption choice with intolerance for declining standard of living. J. Math. Econom. 45, 449–464.CrossRefGoogle Scholar
Riedel, F. and Su, X. (2011). On irreversible investment. Finance Stoch. 15, 607–633.CrossRefGoogle Scholar
Watson, J. G. and Scott, J. S. (2014). Ratchet consumption over finite and infinite planning horizons. J. Math. Econom. 54, 84–96.CrossRefGoogle Scholar