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Forward backward SDEs in weak formulation
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 |
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.
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. |
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. |
[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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