A partial information non-zero sum differential game of backward stochastic differential equations with applications. (English) Zbl 1260.93181

Summary: This paper is concerned with a new kind of non-zero sum differential game of Backward Stochastic Differential Equations (BSDEs). It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motion. We establish a necessary condition in the form of Pontryagin’s maximum principle for open-loop Nash equilibrium point of this type of partial information game, and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a partial information Linear-Quadratic (LQ) game and a partial information financial problem.


93E20 Optimal stochastic control
49K45 Optimality conditions for problems involving randomness
91A23 Differential games (aspects of game theory)
91A15 Stochastic games, stochastic differential games
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI


[1] An, T.T.K.; Øksendal, B., Maximum principal for stochastic differential games with partial information, Journal of optimization theory and applications, 139, 463-483, (2008) · Zbl 1159.91321
[2] Başar, T.; Olsder, G.J., Dynamic noncooperative game theory, (1982), Academic Press London · Zbl 0479.90085
[3] El Karoui, N.; Hamadène, S., BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations, Stochastic processes and their applications, 107, 145-169, (2003) · Zbl 1075.60534
[4] Hamadène, S., Nonzero-sum linear-quadratic stochastic differential games and backward-forward equations, Stochastic analysis and applications, 17, 117-130, (1999) · Zbl 0922.60050
[5] Hamadène, S., Mixed zero-sum stochastic differential game and American game options, SIAM journal on control and optimization, 45, 496-518, (2006) · Zbl 1223.91009
[6] Hamadène, S.; Lepeltier, J.P.; Peng, S., BSDEs with continuous coefficients and application to Markovian nonzero sum stochastic differential games, (), 115-128 · Zbl 0892.60062
[7] Huang, J.; Wang, G.; Xiong, J., A maximum principle for partial information backward stochastic control problems with applications, SIAM journal on control and optimization, 48, 2106-2117, (2009) · Zbl 1203.49037
[8] Hu, Y.; Peng, S., Solution of forward – backward stochastic differential equations, Probability theory and related fields, 103, 273-283, (1995) · Zbl 0831.60065
[9] Lim, A.E.B.; Zhou, X., Linear-quadratic control of backward stochastic differential equations, SIAM journal on control and optimization, 40, 450-474, (2001) · Zbl 0995.93074
[10] Pardoux, E.; Peng, S., Adapted solutions of a backward stochastic differential equation, Systems and control letters, 14, 55-61, (1990) · Zbl 0692.93064
[11] Peng, S., Backward stochastic differential equations and applications to optimal control, Applied mathematics and optimization, 27, 125-144, (1993) · Zbl 0769.60054
[12] Peng, S.; Wu, Z., Fully coupled forward – backward stochastic differential equations and applications to the optimal control, SIAM journal on control and optimization, 37, 825-843, (1999) · Zbl 0931.60048
[13] Pham, H., Mean-variance hedging for partially observed drift processes, International journal of theoretical and applied finance, 4, 263-284, (2001) · Zbl 1153.91554
[14] Song, Q.; Yin, G.; Zhang, Z., Numerical solutions for stochastic differential games with regime switching, IEEE transactions on automatic control, 53, 509-521, (2008) · Zbl 1367.91028
[15] Wang, G.; Wu, Z., Kalman-bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems, Journal of mathematical analysis and applications, 342, 1280-1296, (2008) · Zbl 1141.93070
[16] Wang, G.; Yu, Z., A pontryagin’s maximum principle for non-zero sum differential games of BSDEs with applications, IEEE transactions on automatic control, 55, 1742-1747, (2010) · Zbl 1368.91035
[17] Wonham, W.M., On the separation theorem of stochastic control, SIAM journal on control, 6, 312-326, (1968) · Zbl 0164.19101
[18] Wu, Z., Forward-backward stochastic differential equations, linear quadratic stochastic optimal control and nonzero sum differential games, Journal of systems science and complexity, 18, 179-192, (2005) · Zbl 1156.93409
[19] Wu, Z., A maximum principle for partially observed optimal control of forward – backward stochastic control systems, Science in China serias F: information sciences, 53, 2205-2214, (2010) · Zbl 1227.93116
[20] Xiong, J., An introduction to stochastic filtering theory, (2008), Oxford University Press London · Zbl 1144.93003
[21] Xiong, J.; Zhou, X., Mean-variance portfolio selection under partial information, SIAM journal on control and optimization, 46, 156-175, (2007) · Zbl 1142.91007
[22] Yong, J.; Zhou, X., Stochastic controls: Hamiltonian systems and HJB euqations, (1999), Springer-Verlag New York
[23] Yu, Z.; Ji, S., Linear-quadratic non-zero sum differential game of backward stochstic differential equations, (), 562-566, July 16-18
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.