# American Institute of Mathematical Sciences

June  2019, 9(2): 257-276. doi: 10.3934/mcrf.2019013

## A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance

 1 Department of Mathematics, Southern University of Science and Technology, Shenzhen, China 2 Department of Mathematics, University of Macau, Macau, China 3 School of Economics and Commerce, Guangdong University of Technology, Guangzhou 510520, China 4 China Wealth (Asset) Management Registry & Custody Co. Ltd, Beijing 100045, China 5 School of Mathematics, Shandong University, Jinan 250100, China

* Corresponding author: Yi Zhuang

Received  February 2017 Revised  February 2018 Published  November 2018

In this article, we study a class of partially observed non-zero sum stochastic differential game based on forward and backward stochastic differential equations (FBSDEs). It is required that each player has his own observation equation, and the corresponding Nash equilibrium control is required to be adapted to the filtration generated by the observation process. To find the Nash equilibrium point, we establish the maximum principle as a necessary condition and derive the verification theorem as a sufficient condition. Applying the theoretical results and stochastic filtering theory, we obtain the explicit investment strategy of a partial information financial problem.

Citation: Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control and Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013
##### References:
 [1] T. T. K. An and B. Øksendal, Maximum principal for stochastic differential games with partial information, Journal of Optimization Theory and Applications, 139 (2008), 463-483.  doi: 10.1007/s10957-008-9398-y. [2] J. Baras, R. Elliott and M. Kohlmann, The partially observed stochastic minimum principle, SIAM J. Control Optim., 27 (1989), 1279-1292.  doi: 10.1137/0327065. [3] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, U. K. 1992. doi: 10.1017/CBO9780511526503. [4] J. Campbell, G. Chacko, J. Rodriguez and L. Viceira, Strategic asset allocation in a continuous-time VAR model, Journal of Economic Dynamics and Control, 28 (2004), 2195-2214.  doi: 10.1016/j.jedc.2003.09.005. [5] N. El Karoui and S. Hamadène, BSDEs and risk-sensitive control, zero-sum and non zero-sum game problems of stochastic functional differential equations, Stochastic Processes and their Applications, 107 (2003), 145-169.  doi: 10.1016/S0304-4149(03)00059-0. [6] S. Hamadène, Non zero-sum linear-quadratic stochastic differential games and backward forward equations, Stochastic Analysis and Applications, 17 (1999), 117-130.  doi: 10.1080/07362999908809591. [7] U. Haussmann, The maximum principle for optimal control of diffusions with partial information, SIAM Journal on Control and Optimization, 25 (1987), 341-361. [8] M. Hu, Stochastic global maximum principle for optimization with recursive utilities, Probability, Uncertainty and Quantitative Risk, 2 (2017), Paper No. 1, 20 pp. doi: 10.1186/s41546-017-0014-7. [9] J. Huang, G. Wang and J. Xiong, A maximum principle for partial information backward stochastic control problems with applications, SIAM J. Control Optim., 48 (2009), 2106-2117.  doi: 10.1137/080738465. [10] E. Hui and H. Xiao, Maximum principle for differential games of forward-backward stochastic systems with applications, J. Math. Anal. Appl., 386 (2012), 412-427.  doi: 10.1016/j.jmaa.2011.08.009. [11] H. Liu, Robust consumption and portfolio choice for time varying investment opportunities, Annals of Finance, 6 (2010), 435-454. [12] J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and Their Applications, Springer-Verlag, New York, 1999. [13] R. Merton, On estimating the expected return on the market: An exploratory investigation, Journal of Financial Economics, 8 (1980), 323-361. [14] J. Nash, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences, 36 (1950), 48-49.  doi: 10.1073/pnas.36.1.48. [15] T. Nie, J. Shi and Z. Wu, Connection between MP and DPP for stochastic recursive optimal control problems: viscosity solution framework in the general case, SIAM J. Control Optim., 55 (2017), 3258-3294.  doi: 10.1137/16M1064957. [16] B. Øksendal and A. Sulem, Forward-backward stochastic differential games and stochastic control under model uncertainty, Journal of Optimization Theory and Applications, 161 (2012), 22-55.  doi: 10.1007/s10957-012-0166-7. [17] E. Pardoux and S. Peng, Adapted solution of backward stochastic differential equation, Syst. Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6. [18] S. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27 (1993), 125-144.  doi: 10.1007/BF01195978. [19] J. Shi and Z. Wu, The maximum principle for partially observed optimal control of fully coupled forward-backward stochastic system, J. Optim. Theory Appl., 145 (2010), 543-578.  doi: 10.1007/s10957-010-9696-z. [20] J. Shi and Z. Wu, Maximum principle for forward-backward stochastic control system with random jumps and applications to finance, Journal of Systems Science & Complexity, 23 (2010), 219-231.  doi: 10.1007/s11424-010-7224-8. [21] M. Tang and Q. Meng, Stochastic differential games of fully coupled forward-backward stochastic systems under partial information, in Proceddings of 29th Chinese Control Conference, Beijing, China, (2010), 1150-1155. [22] J. Von Neumann and O. Morgenstern, The Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944. [23] G. Wang and Z. Wu, 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.  doi: 10.1016/j.jmaa.2007.12.072. [24] G. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information, IEEE Trans. Automat. Control, 54 (2009), 1230-1242.  doi: 10.1109/TAC.2009.2019794. [25] G. Wang, Z. Wu and J. Xiong, Maximum principles for forward-backward stochastic control systems with correlated state and obervation noises, SIAM J. Control Optim., 51 (2013), 491-524.  doi: 10.1137/110846920. [26] G. Wang and Z. Yu, A Pontryagin's maximum principle for non-zero sum differential games of BSDEs with applications, IEEE Transactions on Automatic Control, 55 (2010), 1742-1747.  doi: 10.1109/TAC.2010.2048052. [27] G. Wang and Z. Yu, A partial information non-zero sum differential game of backward stochastic differential equations with applications, Automatica, 48 (2012), 342-352.  doi: 10.1016/j.automatica.2011.11.010. [28] Z. Wu, Forward-backward stochastic differential equations, linear quadratic stochastic optimal control and nonzero sum differential games, Journal of Systems Science and Complexity, 18 (2005), 179-192. [29] Z. Wu, A maximum principle for partially observed optimal control of forward-backward stochastic control systems, Sci. China Ser. F Inf. Sci., 53 (2010), 2205-2214.  doi: 10.1007/s11432-010-4094-6. [30] Z. Wu, A general maximum principle for optimal control problems of forward-backward stochastic control systems, Automatica, 49 (2013), 1473-1480.  doi: 10.1016/j.automatica.2013.02.005. [31] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, Oxford, 2008. [32] J. Xiong and X. Zhou, Mean-variance portfolio selection under partial information, SIAM Journal on Control and Optimization, 46 (2007), 156-175.  doi: 10.1137/050641132. [33] W. Xu, Stochastic maximum principle for optimal control problem of forward and backward system, J. Aust. Math. Soc. B, 37 (1995), 172-185.  doi: 10.1017/S0334270000007645. [34] J. Yong, Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions, SIAM Journal on Control and Optimization, 48 (2010), 4119-4156.  doi: 10.1137/090763287. [35] J. Yong and X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3. [36] Z. Yu and S. Ji, Linear-quadratic non-zero sum differential game of backward stochastic differential equations, in Proceddings 27th Chinese Control Conference, Kunming, Yunnan, (2008), 562-566.

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##### References:
 [1] T. T. K. An and B. Øksendal, Maximum principal for stochastic differential games with partial information, Journal of Optimization Theory and Applications, 139 (2008), 463-483.  doi: 10.1007/s10957-008-9398-y. [2] J. Baras, R. Elliott and M. Kohlmann, The partially observed stochastic minimum principle, SIAM J. Control Optim., 27 (1989), 1279-1292.  doi: 10.1137/0327065. [3] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, U. K. 1992. doi: 10.1017/CBO9780511526503. [4] J. Campbell, G. Chacko, J. Rodriguez and L. Viceira, Strategic asset allocation in a continuous-time VAR model, Journal of Economic Dynamics and Control, 28 (2004), 2195-2214.  doi: 10.1016/j.jedc.2003.09.005. [5] N. El Karoui and S. Hamadène, BSDEs and risk-sensitive control, zero-sum and non zero-sum game problems of stochastic functional differential equations, Stochastic Processes and their Applications, 107 (2003), 145-169.  doi: 10.1016/S0304-4149(03)00059-0. [6] S. Hamadène, Non zero-sum linear-quadratic stochastic differential games and backward forward equations, Stochastic Analysis and Applications, 17 (1999), 117-130.  doi: 10.1080/07362999908809591. [7] U. Haussmann, The maximum principle for optimal control of diffusions with partial information, SIAM Journal on Control and Optimization, 25 (1987), 341-361. [8] M. Hu, Stochastic global maximum principle for optimization with recursive utilities, Probability, Uncertainty and Quantitative Risk, 2 (2017), Paper No. 1, 20 pp. doi: 10.1186/s41546-017-0014-7. [9] J. Huang, G. Wang and J. Xiong, A maximum principle for partial information backward stochastic control problems with applications, SIAM J. Control Optim., 48 (2009), 2106-2117.  doi: 10.1137/080738465. [10] E. Hui and H. Xiao, Maximum principle for differential games of forward-backward stochastic systems with applications, J. Math. Anal. Appl., 386 (2012), 412-427.  doi: 10.1016/j.jmaa.2011.08.009. [11] H. Liu, Robust consumption and portfolio choice for time varying investment opportunities, Annals of Finance, 6 (2010), 435-454. [12] J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and Their Applications, Springer-Verlag, New York, 1999. [13] R. Merton, On estimating the expected return on the market: An exploratory investigation, Journal of Financial Economics, 8 (1980), 323-361. [14] J. Nash, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences, 36 (1950), 48-49.  doi: 10.1073/pnas.36.1.48. [15] T. Nie, J. Shi and Z. Wu, Connection between MP and DPP for stochastic recursive optimal control problems: viscosity solution framework in the general case, SIAM J. Control Optim., 55 (2017), 3258-3294.  doi: 10.1137/16M1064957. [16] B. Øksendal and A. Sulem, Forward-backward stochastic differential games and stochastic control under model uncertainty, Journal of Optimization Theory and Applications, 161 (2012), 22-55.  doi: 10.1007/s10957-012-0166-7. [17] E. Pardoux and S. Peng, Adapted solution of backward stochastic differential equation, Syst. Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6. [18] S. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27 (1993), 125-144.  doi: 10.1007/BF01195978. [19] J. Shi and Z. Wu, The maximum principle for partially observed optimal control of fully coupled forward-backward stochastic system, J. Optim. Theory Appl., 145 (2010), 543-578.  doi: 10.1007/s10957-010-9696-z. [20] J. Shi and Z. Wu, Maximum principle for forward-backward stochastic control system with random jumps and applications to finance, Journal of Systems Science & Complexity, 23 (2010), 219-231.  doi: 10.1007/s11424-010-7224-8. [21] M. Tang and Q. Meng, Stochastic differential games of fully coupled forward-backward stochastic systems under partial information, in Proceddings of 29th Chinese Control Conference, Beijing, China, (2010), 1150-1155. [22] J. Von Neumann and O. Morgenstern, The Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944. [23] G. Wang and Z. Wu, 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.  doi: 10.1016/j.jmaa.2007.12.072. [24] G. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information, IEEE Trans. Automat. Control, 54 (2009), 1230-1242.  doi: 10.1109/TAC.2009.2019794. [25] G. Wang, Z. Wu and J. Xiong, Maximum principles for forward-backward stochastic control systems with correlated state and obervation noises, SIAM J. Control Optim., 51 (2013), 491-524.  doi: 10.1137/110846920. [26] G. Wang and Z. Yu, A Pontryagin's maximum principle for non-zero sum differential games of BSDEs with applications, IEEE Transactions on Automatic Control, 55 (2010), 1742-1747.  doi: 10.1109/TAC.2010.2048052. [27] G. Wang and Z. Yu, A partial information non-zero sum differential game of backward stochastic differential equations with applications, Automatica, 48 (2012), 342-352.  doi: 10.1016/j.automatica.2011.11.010. [28] Z. Wu, Forward-backward stochastic differential equations, linear quadratic stochastic optimal control and nonzero sum differential games, Journal of Systems Science and Complexity, 18 (2005), 179-192. [29] Z. Wu, A maximum principle for partially observed optimal control of forward-backward stochastic control systems, Sci. China Ser. F Inf. Sci., 53 (2010), 2205-2214.  doi: 10.1007/s11432-010-4094-6. [30] Z. Wu, A general maximum principle for optimal control problems of forward-backward stochastic control systems, Automatica, 49 (2013), 1473-1480.  doi: 10.1016/j.automatica.2013.02.005. [31] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, Oxford, 2008. [32] J. Xiong and X. Zhou, Mean-variance portfolio selection under partial information, SIAM Journal on Control and Optimization, 46 (2007), 156-175.  doi: 10.1137/050641132. [33] W. Xu, Stochastic maximum principle for optimal control problem of forward and backward system, J. Aust. Math. Soc. B, 37 (1995), 172-185.  doi: 10.1017/S0334270000007645. [34] J. Yong, Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions, SIAM Journal on Control and Optimization, 48 (2010), 4119-4156.  doi: 10.1137/090763287. [35] J. Yong and X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3. [36] Z. Yu and S. Ji, Linear-quadratic non-zero sum differential game of backward stochastic differential equations, in Proceddings 27th Chinese Control Conference, Kunming, Yunnan, (2008), 562-566.
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