Time-inconsistent recursive zero-sum stochastic differential games

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September 2018, 8(3&4): 1051-1079. doi: 10.3934/mcrf.2018045

Time-inconsistent recursive zero-sum stochastic differential games

1.	School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
2.	School of Mathematics, Shandong University, Jinan 250100, China

* Corresponding author: Zhiyong Yu

Dedicated to Professor Jiongmin Yong’s 60 Birthday

Received August 2017 Revised March 2018 Published September 2018

Fund Project: This work is supported in part by the National Natural Science Foundation of China (11471192, 11401091, 11571203), the Nature Science Foundation of Shandong Province (JQ201401), the Fundamental Research Funds of Shandong University (2017JC016), and the Fundamental Research Funds for the Central Universities (2412017FZ008)

In this paper, a kind of time-inconsistent recursive zero-sum stochastic differential game problems are studied by a hierarchical backward sequence of time-consistent subgames. The notion of feedback control-strategy law is adopted to constitute a closed-loop formulation. Instead of the time-inconsistent saddle points, a new concept named equilibrium saddle points is introduced and investigated, which is time-consistent and can be regarded as a local approximate saddle point in a proper sense. Moreover, a couple of equilibrium Hamilton-Jacobi-Bellman-Isaacs equations are obtained to characterize the equilibrium values and construct the equilibrium saddle points.

Keywords: Time-inconsistency, stochastic differential game, equilibrium saddle point, equilibrium Hamilton-Jacobi-Bellman-Isaacs equation, recursive criterion functional.

Mathematics Subject Classification: Primary: 49N70; Secondary: 60H10.

Citation: Qingmeng Wei, Zhiyong Yu. Time-inconsistent recursive zero-sum stochastic differential games. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1051-1079. doi: 10.3934/mcrf.2018045

References:

[1]	T. Björk and A. Murgoci, A theory of Markovian time-inconsistent stochastic control in discrete time, Finance Stoch., 18 (2014), 545-592. doi: 10.1007/s00780-014-0234-y.
[2]	T. Björk, A. Murgoci and X. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24. doi: 10.1111/j.1467-9965.2011.00515.x.
[3]	R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamiltion-Jacobi-Bellman-Isaacs equation, SIAM J. Control Optim., 47 (2008), 444-475. doi: 10.1137/060671954.
[4]	D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394. doi: 10.2307/2951600.
[5]	I. Ekeland and A. Lazrak, The golden rule when preferences are time inconsistent, Math. Financ. Econ., 4 (2010), 29-55. doi: 10.1007/s11579-010-0034-x.
[6]	N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.
[7]	R. Elliott and N. J. Kalton, Values in differential games, Bulletin of the American Mathematical Society, 78 (1972), 427-431. doi: 10.1090/S0002-9904-1972-12929-X.
[8]	L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana Univ. Math. J., 33 (1984), 773-797. doi: 10.1512/iumj.1984.33.33040.
[9]	W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana Univ. Math. J., 38 (1989), 293-314. doi: 10.1512/iumj.1989.38.38015.
[10]	Y. Hu, H. Jin and X. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572. doi: 10.1137/110853960.
[11]	J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly - a four step scheme, Probab. Theory Related Fields, 98 (1994), 339-359. doi: 10.1007/BF01192258.
[12]	E. Pardoux and S. Peng, Backward stochastic differential equations and quasi-linear parabolic partial differential equations, in: Stochastic Partial Differential Equations and their Applications. Lect. Notes in Control & Info. Sci. (eds. B. L. Rozovskii and R. S. Sowers), 176, Springer, Berlin, Heidelberg, 1992,200-217. doi: 10.1007/BFb0007334.
[13]	S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., 37 (1991), 61-74. doi: 10.1080/17442509108833727.
[14]	R. A. Pollak, Consistent planning, Review of Economic Studies, 35 (1968), 201-208. doi: 10.2307/2296548.
[15]	Q. Wei, J. Yong and Z. Yu, Time-inconsistent recursive stochastic optimal control problems, SIAM J. Control Optim., 55 (2017), 4156-4201. doi: 10.1137/16M1079415.
[16]	J. Yong, A deterministic linear quadratic time-inconsistent optimal control problem, Math. Control Relat. Fields, 1 (2011), 83-118. doi: 10.3934/mcrf.2011.1.83.
[17]	J. Yong, Time-inconsistent optimal control problems and the equilibrium HJB equation, Math. Control Relat. Fields, 2 (2012), 271-329. doi: 10.3934/mcrf.2012.2.271.
[18]	J. Yong, Time-inconsistent optimal control problems, Proceedings of 2014 ICM, Section 16. Control Theory and Optimization, 4 (2014), 947-969.
[19]	J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations — time-consistent solutions, Trans. Amer. Math. Soc., 369 (2017), 5467-5523. doi: 10.1090/tran/6502.
[20]	J. Yong and X. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations, Applications of Mathematics (New York), 43, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.
[21]	Z. Yu, An optimal feedback control-strategy pair for zero-sum linear-quadratic stochastic differential game: The Riccati equation approach, SIAM J. Control Optim., 53 (2015), 2141-2167. doi: 10.1137/130947465.

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References:

[1]	T. Björk and A. Murgoci, A theory of Markovian time-inconsistent stochastic control in discrete time, Finance Stoch., 18 (2014), 545-592. doi: 10.1007/s00780-014-0234-y.
[2]	T. Björk, A. Murgoci and X. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24. doi: 10.1111/j.1467-9965.2011.00515.x.
[3]	R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamiltion-Jacobi-Bellman-Isaacs equation, SIAM J. Control Optim., 47 (2008), 444-475. doi: 10.1137/060671954.
[4]	D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394. doi: 10.2307/2951600.
[5]	I. Ekeland and A. Lazrak, The golden rule when preferences are time inconsistent, Math. Financ. Econ., 4 (2010), 29-55. doi: 10.1007/s11579-010-0034-x.
[6]	N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.
[7]	R. Elliott and N. J. Kalton, Values in differential games, Bulletin of the American Mathematical Society, 78 (1972), 427-431. doi: 10.1090/S0002-9904-1972-12929-X.
[8]	L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana Univ. Math. J., 33 (1984), 773-797. doi: 10.1512/iumj.1984.33.33040.
[9]	W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana Univ. Math. J., 38 (1989), 293-314. doi: 10.1512/iumj.1989.38.38015.
[10]	Y. Hu, H. Jin and X. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572. doi: 10.1137/110853960.
[11]	J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly - a four step scheme, Probab. Theory Related Fields, 98 (1994), 339-359. doi: 10.1007/BF01192258.
[12]	E. Pardoux and S. Peng, Backward stochastic differential equations and quasi-linear parabolic partial differential equations, in: Stochastic Partial Differential Equations and their Applications. Lect. Notes in Control & Info. Sci. (eds. B. L. Rozovskii and R. S. Sowers), 176, Springer, Berlin, Heidelberg, 1992,200-217. doi: 10.1007/BFb0007334.
[13]	S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., 37 (1991), 61-74. doi: 10.1080/17442509108833727.
[14]	R. A. Pollak, Consistent planning, Review of Economic Studies, 35 (1968), 201-208. doi: 10.2307/2296548.
[15]	Q. Wei, J. Yong and Z. Yu, Time-inconsistent recursive stochastic optimal control problems, SIAM J. Control Optim., 55 (2017), 4156-4201. doi: 10.1137/16M1079415.
[16]	J. Yong, A deterministic linear quadratic time-inconsistent optimal control problem, Math. Control Relat. Fields, 1 (2011), 83-118. doi: 10.3934/mcrf.2011.1.83.
[17]	J. Yong, Time-inconsistent optimal control problems and the equilibrium HJB equation, Math. Control Relat. Fields, 2 (2012), 271-329. doi: 10.3934/mcrf.2012.2.271.
[18]	J. Yong, Time-inconsistent optimal control problems, Proceedings of 2014 ICM, Section 16. Control Theory and Optimization, 4 (2014), 947-969.
[19]	J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations — time-consistent solutions, Trans. Amer. Math. Soc., 369 (2017), 5467-5523. doi: 10.1090/tran/6502.
[20]	J. Yong and X. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations, Applications of Mathematics (New York), 43, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.
[21]	Z. Yu, An optimal feedback control-strategy pair for zero-sum linear-quadratic stochastic differential game: The Riccati equation approach, SIAM J. Control Optim., 53 (2015), 2141-2167. doi: 10.1137/130947465.

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