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Equilibrium stochastic control with implicitly defined objective functions

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  • Zongxia Liang
  • Jianming Xia
  • Keyu Zhang

Abstract

This paper considers a class of stochastic control problems with implicitly defined objective functions, which are the sources of time-inconsistency. We study the closed-loop equilibrium solutions in a general controlled diffusion framework. First, we provide a sufficient and necessary condition for a strategy to be an equilibrium. Then, we apply the result to discuss two problems of dynamic portfolio selection for a class of betweenness preferences, allowing for closed convex constraints on portfolio weights and borrowing cost, respectively. The equilibrium portfolio strategies are explicitly characterized in terms of the solutions of some first-order ordinary differential equations for the case of deterministic market coefficients.

Suggested Citation

  • Zongxia Liang & Jianming Xia & Keyu Zhang, 2023. "Equilibrium stochastic control with implicitly defined objective functions," Papers 2312.15173, arXiv.org, revised Dec 2023.
    • Handle: RePEc:arx:papers:2312.15173
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    References listed on IDEAS

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    1. Kuan, Chung-Ming & Yeh, Jin-Huei & Hsu, Yu-Chin, 2009. "Assessing value at risk with CARE, the Conditional Autoregressive Expectile models," Journal of Econometrics, Elsevier, vol. 150(2), pages 261-270, June.
    2. Ying Hu & Hanqing Jin & Xun Yu Zhou, 2012. "Time-Inconsistent Stochastic Linear--Quadratic Control," Post-Print hal-00691816, HAL.
    3. Robert J. Aumann & Roberto Serrano, 2008. "An Economic Index of Riskiness," Journal of Political Economy, University of Chicago Press, vol. 116(5), pages 810-836, October.
    4. Zongxia Liang & Jianming Xia & Fengyi Yuan, 2023. "Dynamic portfolio selection for nonlinear law-dependent preferences," Papers 2311.06745, arXiv.org, revised Nov 2023.
    5. Tiantian Mao & Jun Cai, 2018. "Risk measures based on behavioural economics theory," Finance and Stochastics, Springer, vol. 22(2), pages 367-393, April.
    6. Dean P. Foster & Sergiu Hart, 2009. "An Operational Measure of Riskiness," Journal of Political Economy, University of Chicago Press, vol. 117(5), pages 785-814.
    7. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    8. Wendell H. Fleming & Thaleia Zariphopoulou, 1991. "An Optimal Investment/Consumption Model with Borrowing," Mathematics of Operations Research, INFORMS, vol. 16(4), pages 802-822, November.
    9. Tomas Björk & Mariana Khapko & Agatha Murgoci, 2017. "On time-inconsistent stochastic control in continuous time," Finance and Stochastics, Springer, vol. 21(2), pages 331-360, April.
    10. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    11. Cuoco, Domenico, 1997. "Optimal Consumption and Equilibrium Prices with Portfolio Constraints and Stochastic Income," Journal of Economic Theory, Elsevier, vol. 72(1), pages 33-73, January.
    12. R. H. Strotz, 1955. "Myopia and Inconsistency in Dynamic Utility Maximization," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 23(3), pages 165-180.
    13. Chew, Soo Hong, 1983. "A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox," Econometrica, Econometric Society, vol. 51(4), pages 1065-1092, July.
    14. Chonghu Guan & Xiaomin Shi & Zuo Quan Xu, 2022. "Continuous-time Markowitz's mean-variance model under different borrowing and saving rates," Papers 2201.00914, arXiv.org, revised May 2023.
    15. Bellini, Fabio & Klar, Bernhard & Müller, Alfred & Rosazza Gianin, Emanuela, 2014. "Generalized quantiles as risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 41-48.
    16. Back, Kerry E., 2017. "Asset Pricing and Portfolio Choice Theory," OUP Catalogue, Oxford University Press, number 9780190241148.
    17. Chonghu Guan & Xiaomin Shi & Zuo Quan Xu, 2023. "Continuous-Time Markowitz’s Mean-Variance Model Under Different Borrowing and Saving Rates," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 167-208, October.
    18. Dekel, Eddie, 1986. "An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom," Journal of Economic Theory, Elsevier, vol. 40(2), pages 304-318, December.
    19. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
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