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Keywords:
time inconsistency; equilibrium strategy; extended HJB equations; mean-variance criterion; investment problem with delay
time inconsistency; equilibrium strategy; extended HJB equations; mean-variance criterion; investment problem with delay
Summary:
This paper develops a game-theoretic framework for analyzing stochastic differential delayed equations (SDDEs) in time-inconsistent control problems. By extending the Bellman equation to a system of nonlinear equations, the framework identifies subgame-perfect Nash equilibrium strategies for delayed processes with functional objectives. The approach accounts for the challenges introduced by delays and time inconsistency, providing a robust method for deriving equilibrium strategies. To illustrate its applicability, the framework is applied to a mean-variance portfolio selection problem with state-dependent risk aversion and delay, demonstrating how past decisions influence current outcomes. This work advances the theoretical understanding of SDDEs and offers practical insights for applications in finance and related fields.
References:
[1] Alia, I., Chighoub, F., Sohail, A.: A characterization of equilibrium strategies in continuous-time mean-variance problems for insurers. Insurance Math. Econom. 68 (2016), 212-223. DOI
[2] Alia, I., Chighoub, F., Sohail, A.: The maximum principle in time-inconsistent LQ equilibrium. Serdica Math. J. 42 (2016), 103-138. DOI
[3] Björk, T., Murgoci, A.: A general theory of Markovian time-inconsistent stochastic control problems. SSRN Preprint (2008).
[4] Björk, T., Murgoci, A., Zhou, X. Y.: Mean-variance portfolio optimization with state-dependent risk aversion. Math. Finance 24 (2014), 1-24. DOI
[5] Björk, T., Khapko, M., Murgoci, A.: On time-inconsistent stochastic control in continuous time. Finance Stoch. 21 (2017), 331-360. DOI
[6] Chang, M. H., Pang, T., Yang, Y.: A stochastic portfolio optimization model with bounded memory. Math. Oper. Res. 36 (2011), 604-619. DOI
[7] Chen, L., Wu, Z.: Maximum principle for the stochastic optimal control problem with delay and application. Automatica 46 (2010), 1074-1080. DOI
[8] David, D.: Optimal control of stochastic delayed systems with jumps. Preprint (2008).
[9] Du, H., Huang, J., Qin, Y.: A stochastic maximum principle for delayed mean-field stochastic differential equations and its applications. IEEE Trans. Automat. Control 58 (2013), 3212-3217. DOI
[10] Ekeland, I., Lazrak, A.: The golden rule when preferences are time-inconsistent. Math. Finance 18 (2008), 385-399. DOI
[11] Ekeland, I., Pirvu, T. A.: Investment and consumption without commitment. Math. Financ. Econ. 2 (2008), 57-86. DOI
[12] Elsanousi, I., Larssen, B.: Optimal consumption under partial observations for a stochastic system with delay. Preprint, Univ. Oslo (2001).
[13] Fouque, J. P., Zhang, Z.: Deep learning methods for mean field control problems with delay. Front. Appl. Math. Stat. 6 (2020), Article 11. DOI
[14] Han, J., Hu, R.: Recurrent neural networks for stochastic control problems with delay. Math. Control Signals Syst. 33 (2021), 775-795. DOI
[15] Kushner, H. J.: Numerical methods for controlled stochastic delay systems. Springer, New York (2008).
[16] Larssen, B., Risebro, N. H.: When are HJB-equations in stochastic control of delay systems finite dimensional?. Stoch. Anal. Appl. 21 (2003), 643-671. DOI
[17] Mohammed, S. E. A.: Stochastic differential equations with memory: theory, examples and applications. Progr. Probab. 6, Birkhäuser (1996).
[18] Øksendal, B., Sulem, A.: A maximum principle for optimal control of stochastic systems with delay. In: Optimal Control and PDEs, IOS Press (2000), 64-74.
[19] Phelps, E. S., Pollak, R. A.: On second-best national saving and game-equilibrium growth. Rev. Econom. Stud. 35 (1968), 185-199. DOI
[20] Pollak, R.: Consistent planning. Rev. Econom. Stud. 35 (1968), 201-208. DOI
[21] Ramsey, F. P.: A mathematical theory of saving. Econom. J. 38 (1928), 543-559. DOI
[22] Shi, J.: Two different approaches to stochastic recursive optimal control problems with delay and applications. arXiv:1304.6182 (2013).
[23] Shen, Y., Meng, Q., Shi, P.: Maximum principle for mean-field jump-diffusions with delay and applications. Automatica 50 (2014), 1565-1579. DOI
[24] Shen, Y., Zeng, Y.: Optimal investment-reinsurance with delay for mean-variance insurers. Insurance Math. Econom. 57 (2014), 1-12. DOI
[25] Sheng, L.: Optimal time-consistent investment-reinsurance strategy with delay and common shocks. Commun. Statist. Theory Methods (2021), 1-38. DOI
[26] Strotz, R.: Myopia and inconsistency in dynamic utility maximization. Rev. Econom. Stud. 23 (1955), 165-180. DOI 10.2307/2295722
[27] Marín-Solano, J., Navas, J.: Consumption and portfolio rules for time-inconsistent investors. Eur. J. Oper. Res. 201 (2010), 860-872. DOI
[28] Yang, B. Z., He, X. J., Zhu, S. P.: Continuous-time mean-variance-utility portfolio problem and its equilibrium strategy. Optim. 71 (2022), 4213-4241. DOI
[29] Yong, J.: A deterministic linear quadratic time-inconsistent optimal control problem. Math. Control Relat. Fields 1 (2011), 83-118. DOI
[30] Yong, J.: Time-inconsistent optimal control problems and the equilibrium HJB equation. Math. Control Relat. Fields 2 (2012), 271-329. DOI
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