Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T16:55:02.696Z Has data issue: false hasContentIssue false
  • English
  • Français

Differential games of partial information forward-backward doubly SDE and applications∗

Published online by Cambridge University Press:  10 October 2013

Eddie C.M. Hui  and
Hua Xiao
Eddie C.M. Hui
Affiliation:
Department of Building and Real Estate, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, P.R. China. bscmhui@polyu.edu.hk
Hua Xiao
Affiliation:
School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, P.R. China; xiaohua@sdu.edu.cn
Get access

Abstract

This paper addresses a new differential game problem with forward-backward doubly stochastic differential equations. There are two distinguishing features. One is that our game systems are initial coupled, rather than terminal coupled. The other is that the admissible control is required to be adapted to a subset of the information generated by the underlying Brownian motions. We establish a necessary condition and a sufficient condition for an equilibrium point of nonzero-sum games and a saddle point of zero-sum games. To illustrate some possible applications, an example of linear-quadratic nonzero-sum differential games is worked out. Applying stochastic filtering techniques, we obtain an explicit expression of the equilibrium point.

Keywords

Stochastic differential gamepartial informationforward-backward doubly stochastic differential equationequilibrium pointstochastic filtering
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014 , pp. 78 - 94
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Biagini, F. and Øksendal, B., Minimal variance hedging for insider trading. Int. J. Theor. Appl. Finance 9 (2006) 1351–1375. Google Scholar
Buckdahn, R. and Li, J., Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim. 47 (2008) 444–475. Google Scholar
Campi, L., Some results on quadratic hedging with insider trading. Stochastics 77 (2005) 327–348. Google Scholar
Fuhrman, M. and Tessitore, G., Existence of optimal stochastic controls and global solutions of forward-backward stochastic differential equations. SIAM J. Control Optim. 43 (2004) 813–830. Google Scholar
Han, Y., Peng, S. and Wu, Z., Maximum principle for backward doubly stochastic control systems with applications. SIAM J. Control Optim. 48 (2010) 4224–4241. Google Scholar
Huang, J., Wang, G. and Xiong, J., A maximum principle for partial information backward stochastic control problems with applications. SIAM J. Control Optim. 40 (2009) 2106–2117. Google Scholar
Hui, E. and Xiao, H., Maximum principle for differential games of forward-backward stochastic systems with applications. J. Math. Anal. Appl. 386 (2012) 412–427. Google Scholar
S. Liptser and N. Shiryaev, Statistics of Random Processes. Springer-verlag (1977).
J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, in vol. 1702 of Lect. Notes Math., Springer-Verlag (1999).
Øksendal, B. and Sulem, A., Maximum principles for optimal control of forward-backward stochastic differential equations with jumps. SIAM J. Control Optim. 48 (2010) 2945–2976. Google Scholar
Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990) 55–61. Google Scholar
Pardoux, E. and Peng, S., Backward doubly stochastic differential equations and systems of quasilinear parabolic SPDE’s. Probab. Theory Relat. Fields 98 (1994) 209–227. Google Scholar
S. Peng and Y. Shi, A type of time-symmetric forward-backward stochastic differential equations, in vol. 336 of C. R. Acadamic Science Paris, Series I (2003) 773-778.
Wang, G. and Wu, Z., Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems. J. Math. Anal. Appl. 342 (2008) 1280–1296. Google Scholar
Wang, G. and Yu, Z., A Pontryagin’s maximum principle for nonzero-sum differential games of BSDEs with applications. IEEE Trans. Automat. Contr. 55 (2010) 1742–1747. Google Scholar
Wang, G. and Yu, Z., A partial information non-zero sum differential games of backward stochastic differential equations with applications. Automatica 48 (2012) 342–352. Google Scholar
Xiao, H. and Wang, G., A necessary condition of optimal control for initial coupled forward-backward stochastic differential equations with partial information. J. Appl. Math. Comput. 37 (2011) 347–359. Google Scholar
J. Xiong, An introduction to stochastic filtering theory. Oxford University Press (2008).
Yong, J., A stochastic linear quadratic optimal control problem with generalized expectation. Stoch. Anal. Appl. 26 (2008) 1136–1160. Google Scholar
Yong, J., Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim. 48 (2010) 4119–4156. Google Scholar
J. Yong and X. Zhou, Stochastic control: Hamiltonian systems and HJB equations. Springer-Verlag, New York (1999).
Yu, Z., Linear quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system. Asian J. Control 14 (2012) 173–185. Google Scholar
Zhang, L. and Shi, Y., Maximum principle for forward-backward doubly stochastic control systems and applications. ESAIM: COCV 17 (2011) 1174–1197. Google Scholar