American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001
[2]

Jean-Claude Zambrini. On the geometry of the Hamilton-Jacobi-Bellman equation. Journal of Geometric Mechanics, 2009, 1 (3) : 369-387. doi: 10.3934/jgm.2009.1.369
[3]

Daniele Castorina, Annalisa Cesaroni, Luca Rossi. On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1251-1263. doi: 10.3934/cpaa.2016.15.1251
[4]

Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control & Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010
[5]

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295
[6]

Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial & Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161
[7]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
[8]

Martino Bardi, Gabriele Terrone. On the homogenization of some non-coercive Hamilton--Jacobi--Isaacs equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 207-236. doi: 10.3934/cpaa.2013.12.207
[9]

Federica Masiero. Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 223-263. doi: 10.3934/dcds.2012.32.223
[10]

Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026
[11]

Haiyang Wang, Zhen Wu. Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation. Mathematical Control & Related Fields, 2015, 5 (3) : 651-678. doi: 10.3934/mcrf.2015.5.651
[12]

Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933
[13]

Jiongmin Yong. Time-inconsistent optimal control problems and the equilibrium HJB equation. Mathematical Control & Related Fields, 2012, 2 (3) : 271-329. doi: 10.3934/mcrf.2012.2.271
[14]

Antonio Avantaggiati, Paola Loreti, Cristina Pocci. Mixed norms, functional Inequalities, and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1855-1867. doi: 10.3934/dcdsb.2014.19.1855
[15]

Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461
[16]

Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513
[17]

María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207
[18]

Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441
[19]

Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883
[20]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (58)
  • HTML views (289)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences