Equilibrium strategies for time-inconsistent stochastic switching systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 64.

An optimal control problem is considered for a stochastic differential equation containing a state-dependent regime switching, with a recursive cost functional. Due to the non-exponential discounting in the cost functional, the problem is time-inconsistent in general. Therefore, instead of finding a global optimal control (which is not possible), we look for a time-consistent (approximately) locally optimal equilibrium strategy. Such a strategy can be represented through the solution to a system of partial differential equations, called an equilibrium Hamilton–Jacob–Bellman (HJB) equation which is constructed via a sequence of multi-person differential games. A verification theorem is proved and, under proper conditions, the well-posedness of the equilibrium HJB equation is established as well.

DOI : 10.1051/cocv/2018051
Classification : 93E20, 49N70, 60G07
Mots-clés : Stochastic switching diffusion, time-inconsistency, stochastic optimal control, equilibrium strategy, Hamilton–Jacobi–Bellman equation
Mei, Hongwei 1 ; Yong, Jiongmin 1

1 
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Mei, Hongwei; Yong, Jiongmin. Equilibrium strategies for time-inconsistent stochastic switching systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 64. doi : 10.1051/cocv/2018051. http://www.numdam.org/articles/10.1051/cocv/2018051/

[1] T. Björk, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time. Financ. Stoch. 21 (2017) 331–360. | DOI | MR | Zbl

[2] T. Björk and A. Murgoci, A theory of Markovian time-inconsistent stochastic control in discrete time. Financ. Stoch. 18 (2014) 545–592. | DOI | MR | Zbl

[3] T. Björk, A. Murgoci and X.Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion. Math. Financ. 24 (2014) 1–24. | DOI | MR | Zbl

[4] S.N. Cohen and R.J. Elliott, Solutions of backward stochastic differential equations on Markov chains. Commun. Stoch. Anal. 2 (2008) 251–262. | MR | Zbl

[5] C. Donnelly, Sufficient stochastic maximum principle in a regime-switching diffusion model. Appl. Math. Optim. 64 (2011) 155–169. | DOI | MR | Zbl

[6] C. Donnelly and A.J. Heunis, Quadratic risk minimization in a regime-swtching model with portfolio constraints. SIAM J. Control Optim. 50 (2012) 2431–2461. | DOI | MR | Zbl

[7] D. Duffie and L.G. Epstein, Stochastic differential utility. Econometrica 60 (1992) 353–394. | DOI | MR | Zbl

[8] D. Duffie and L.G. Epstein, Asset pricing with stochastic differential utility. Rev. Financ. Stud. 5 (1992) 411–436. | DOI

[9] D. Duffie and P.L. Lions, PDE solutions of stochastic differential utility. J. Math. Econom. 21 (1992) 577–606. | DOI | MR | Zbl

[10] S.D. Eidel’Man, Parabolic Systems. North Holland Publishing Company, Amsterdam (1969). | Zbl

[11] I. Ekeland and A. Lazrak, The golden rule when preferences are time inconsistent. Math. Financ. Econ. 4 (2010) 29–55. | DOI | MR | Zbl

[12] N. El Karoui, S. Peng and M.C. Quenez, Backward stochastic differential equations in finance. Math. Financ. 7 (1997) 1–71. | DOI | MR | Zbl

[13] N.C. Framstad, B.K. ∅Ksendal and A. Sulem, Sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance. J. Optim. Theory Appl. 121 (2004) 77–98. | DOI | MR | Zbl

[14] A. Friedman, Partial Differential Equations of Parabolic Type. Prentice Hall, Inc., Englewood Cliffs, NJ (1964). | MR | Zbl

[15] J.D. Hamilton, A new approach to the economic analysis of nonstationry time series and the business cycle. Econometrica 57 (1989) 357–384. | DOI | MR | Zbl

[16] A.J. Heunis, Utility maximization in a regime switching model wih convex portfolio constraints and margin requirements: optimality relations and explicit solutions. SIAM J. Control Optim. 53 (2015) 2608–2656. | DOI | MR | Zbl

[17] Y. Hu, H. Jin and X.Y. Zhou, Time-inconsistent stochastic linear-quadratic control. SIAM J. Control Optim. 50 (2012) 1548–1572. | DOI | MR | Zbl

[18] D. Hume, A Treatise of Human Nature, 1st edn. (1739); Reprint. Oxford University Press, New York (1978).

[19] C. Karnam, J. Ma and J. Zhang, Dynamic approaches for some time inconsistent problems. Preprint (2016). | arXiv | MR

[20] H. Kraft and F.T. Seifried, Stochastic differential utility as the continuous-time limit of recursive utility. J. Econ. Theory 151 (2014) 528–550. | DOI | MR | Zbl

[21] T. Kruse and A. Popier, BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration. Stochastics 88 (2016) 491–539. | DOI | MR | Zbl

[22] T. Kruse and A. Popier, Lp-solution for BSDEs with jumps in the case p < 2: Corrections to the paper BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration. Stochastics 89 (2017) 1201–1227 | DOI | MR | Zbl

[23] D. Laibson, Golden eggs and hyperbolic discounting. Quart. J. Econ. 112 (1997) 443–477. | DOI | Zbl

[24] A. Lazrak, Generalized stochastic differential utility and preference for information. Ann. Appl. Probab. 14 (2004) 2149–2175. | DOI | MR | Zbl

[25] A. Lazrak and M.C. Quenez, A generalized stochastic differential utility. Math. Oper. Res. 28 (2003) 154–180. | DOI | MR | Zbl

[26] J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly – a four step scheme. Probab. Theory Relat. Fields 98 (1994) 339–359. | DOI | MR | Zbl

[27] J. Ma, Z. Wu, D. Zhang and J. Zhang, On well-posedness of forward-backward SDEs – a unified approach. Ann. Appl. Probab. 25 (2015) 2168–2214. | MR | Zbl

[28] J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications. Vol. 1702 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1999). | MR | Zbl

[29] J. Marin-Solano and J. Navas, Consumption and portfolio rules for time-inconsistent investors. Eur. J. Oper. Res. 201 (2010) 860–872. | DOI | MR | Zbl

[30] J. Marin-Solano and E.V. Shevkoplyas, Non-constant discounting and differential games with random time horizon. Automatica 47 (2011) 2626–2638. | DOI | MR | Zbl

[31] B.K. ∅Ksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions. Springer-Verlag, Berlin (2005). | MR | Zbl

[32] I. Palacios-Huerta, Time-inconsistent preferences in Adam Smith and David Hume. Hist. Political Econ. 35 (2003) 241–268. | DOI

[33] E. Pardoux and S. Peng, Adapted solution of backward stochastic equation. Syst. Control Lett. 14 (1990) 55–61. | DOI | MR | Zbl

[34] R.A. Pollak, Consistent planning. Rev. Econ. Stud. 35 (1968) 185–199. | DOI

[35] R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering. Springer Science & Business Media, New York (2006). | MR | Zbl

[36] A. Smith, The Theory of Moral Sentiments, 1st edn. (1759). Reprint, Oxford University Press, New York (1976).

[37] L.R. Sotomayor and A. Cadenilla, Explicit solutions of consumption-investment problems in financecial markets with regime switching. Math. Financ. 19 (2009) 251–279. | DOI | MR | Zbl

[38] R.H. Strotz, Myopia and inconsistency in dynamic utility maximization. Rev. Econ. Stud. 23 (1955) 165–180. | DOI

[39] J. Wei, Time-inconsistent optimal control problems with regime-switching. Math. Control Relat. Fields 7 (2017) 585–622. | DOI | MR | Zbl

[40] Q. Wei, J. Yong and Z. Yu, Time-inconsistent recursive stochastic optimal control problems. SIAM J. Control Optim. 55 (2017) 4156–4201. | DOI | MR | Zbl

[41] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Vol. 63. Springer Science & Business Media, New York (2009). | MR | Zbl

[42] J. Yong, Time-inconsistent optimal control problems and the equilibrium HJB equation. Math. Control Relat. Fields 2 (2012) 271–329. | DOI | MR | Zbl

[43] J. Yong, Time-inconsistent optimal control problems, in Proceedings of the International Congress of Mathematicians (2014) 947–969. | MR | Zbl

[44] J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations – time-consistent solutions. Trans. AMS 369 (2017) 5467–5523. | DOI | MR | Zbl

[45] J. Yong and X.Y. Zhou, Stochastic Control: Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999). | DOI | MR | Zbl

[46] J. Zhang, Backward Stochastic Differential Equations – From Linear to Fully Nonlinear Theory. Springer, New York (2017). | DOI | MR

[47] X. Zhang, R.J. Elliott and T.K. Siu, A stochastic maximum principle for a Markov regime-swtching jump-diffusion model and its application to finance. SIAM J. Control Optim. 50 (2012) 964–990. | DOI | MR | Zbl

[48] X.Y. Zhou and G. Yin, Markowitz’s mean-varaince portfolio selection with regime swtching: a contionus-time model. SIAM J. Control Optim. 42 (2003) 1466–1482. | DOI | MR | Zbl

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