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

Time-inconsistent recursive zero-sum stochastic differential games

Qingmeng Wei 1, and  Zhiyong Yu 2,,

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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394.  doi: 10.2307/2951600.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., 37 (1991), 61-74.  doi: 10.1080/17442509108833727.  Google Scholar

[14]

R. A. Pollak, Consistent planning, Review of Economic Studies, 35 (1968), 201-208.  doi: 10.2307/2296548.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Yong, Time-inconsistent optimal control problems, Proceedings of 2014 ICM, Section 16. Control Theory and Optimization, 4 (2014), 947-969.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394.  doi: 10.2307/2951600.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., 37 (1991), 61-74.  doi: 10.1080/17442509108833727.  Google Scholar

[14]

R. A. Pollak, Consistent planning, Review of Economic Studies, 35 (1968), 201-208.  doi: 10.2307/2296548.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Yong, Time-inconsistent optimal control problems, Proceedings of 2014 ICM, Section 16. Control Theory and Optimization, 4 (2014), 947-969.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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