-
Previous Article
On a logarithmic stability estimate for an inverse heat conduction problem
- MCRF Home
- This Issue
-
Next Article
Construction of the minimum time function for linear systems via higher-order set-valued methods
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 |
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.
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. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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. Google Scholar |
[1] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[2] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[3] |
Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 |
[4] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[5] |
J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 |
[6] |
Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208 |
[7] |
María J. Garrido-Atienza, Bohdan Maslowski, Jana Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 |
[8] |
Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023 |
[9] |
Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201 |
[10] |
Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021008 |
[11] |
Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021004 |
[12] |
Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113 |
[13] |
Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 |
[14] |
Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207 |
[15] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[16] |
Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181 |
[17] |
Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 |
[18] |
Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 |
[19] |
Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 |
[20] |
Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 |
2019 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]