American Institute of Mathematical Sciences

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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

Jie Xiong 1,2, , Shuaiqi Zhang 3, and  Yi Zhuang 4,5,,

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.

Keywords: Forward-backward stochastic differential equation, stochastic differential game, maximum principle, equilibrium point, stochastic filtering.
Mathematics Subject Classification: Primary: 49N70; Secondary: 93C20, 93E11.
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 & 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. Google Scholar

[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. Google Scholar

[3]

A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, U. K. 1992. doi: 10.1017/CBO9780511526503. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

U. Haussmann, The maximum principle for optimal control of diffusions with partial information, SIAM Journal on Control and Optimization, 25 (1987), 341-361. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

H. Liu, Robust consumption and portfolio choice for time varying investment opportunities, Annals of Finance, 6 (2010), 435-454. Google Scholar

[12]

J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and Their Applications, Springer-Verlag, New York, 1999. Google Scholar

[13]

R. Merton, On estimating the expected return on the market: An exploratory investigation, Journal of Financial Economics, 8 (1980), 323-361. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

S. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27 (1993), 125-144. doi: 10.1007/BF01195978. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[22]

J. Von Neumann and O. Morgenstern, The Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, Oxford, 2008. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[3]

A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, U. K. 1992. doi: 10.1017/CBO9780511526503. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

U. Haussmann, The maximum principle for optimal control of diffusions with partial information, SIAM Journal on Control and Optimization, 25 (1987), 341-361. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

H. Liu, Robust consumption and portfolio choice for time varying investment opportunities, Annals of Finance, 6 (2010), 435-454. Google Scholar

[12]

J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and Their Applications, Springer-Verlag, New York, 1999. Google Scholar

[13]

R. Merton, On estimating the expected return on the market: An exploratory investigation, Journal of Financial Economics, 8 (1980), 323-361. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

S. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27 (1993), 125-144. doi: 10.1007/BF01195978. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[22]

J. Von Neumann and O. Morgenstern, The Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, Oxford, 2008. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

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