American Institute of Mathematical Sciences

Advanced
  • Previous Article
    An efficient distributed optimization and coordination protocol: Application to the emergency vehicle management
  • JIMO Home
  • This Issue
  • Next Article
    A new approach for uncertain multiobjective programming problem based on $\mathcal{P}_{E}$ principle
January  2015, 11(1): 27-40. doi: 10.3934/jimo.2015.11.27

Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance

Dejian Chang 1, and  Zhen Wu 1,

1. 

School of Mathematics, Shandong University, Jinan 250100, China, China

Received  January 2013 Revised  November 2013 Published  May 2014

This paper is concerned with a maximum principle for a new class of non-zero sum stochastic differential games. Compared with the existing literature, the game systems in this paper are forward-backward systems in which the control variables consist of two components: the continuous controls and the impulse controls. Necessary optimality conditions and sufficient optimality conditions in the form of maximum principle are obtained respectively for open-loop Nash equilibrium point of the foregoing games. A fund management problem is used to shed light on the application of the theoretical results, and the optimal investment portfolio and optimal impulse consumption strategy are obtained explicitly.
Keywords: Non-zero sum stochastic differential game, stochastic recursive utility., maximum principle, open-loop Nash equilibrium point, forward-backward stochastic differential equations, impulse controls.
Mathematics Subject Classification: Primary: 93E20, 91A23; Secondary: 91G8.
Citation: Dejian Chang, Zhen Wu. Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance. Journal of Industrial & Management Optimization, 2015, 11 (1) : 27-40. doi: 10.3934/jimo.2015.11.27
References:
[1]

T. T. K. An and B. Øksendal, Maximum principle 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]

T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, Mathematics in Science and Engineering, 160. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.  Google Scholar

[3]

A. Bensoussan, Lectures on Stochastic Control, in Nonlinear Filtering and Stochastic Control, ser. Lecture Notes in Mathematics, New York: Springer Verlag, 1982.  Google Scholar

[4]

A. Cadenillas and I. Karatzas, The stochastic maximum principle for linear convex optimal control with random coefficients, SIAM J. Control Optim., 33 (1995), 590-624. doi: 10.1137/S0363012992240722.  Google Scholar

[5]

A. Cadenillas and F. Zapatero, Classical and impulse stochastic control of the exchange rate using interest rates and reserves, Math. Finance, 10 (2000), 141-156. doi: 10.1111/1467-9965.00086.  Google Scholar

[6]

M. H. A. Davis and A. Norman, Portfolio selection with transaction costs, Math. Oper. Res., 15 (1990), 676-713. doi: 10.1287/moor.15.4.676.  Google Scholar

[7]

D. Duffie and L. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394. doi: 10.2307/2951600.  Google Scholar

[8]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.  Google Scholar

[9]

S. Hamadéne, Nonzero-sum linear-quadratic stochastic differential games and backward-forward equations, Stochastic Anal. Appl., 17 (1999), 117-130. doi: 10.1080/07362999908809591.  Google Scholar

[10]

E. C. M., 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]

R. Isaacs, Differential Games,, Parts 1-4. The RAND Corporation, (): 1.   Google Scholar

[12]

M. Jeanblanc-Pique, Impulse control method and exchange rate, Math. Finance, 3 (1993), 161-177. doi: 10.1111/j.1467-9965.1993.tb00085.x.  Google Scholar

[13]

R. Korn, Some appliations of impulse control in mathematical finance, Math. Meth. Oper. Res., 50 (1999), 493-518. doi: 10.1007/s001860050083.  Google Scholar

[14]

A. E. B. Lim and X. Zhou, Risk-sensitive control with HARA utility, IEEE Trans. Autom. Control, 46 (2001), 563-578. doi: 10.1109/9.917658.  Google Scholar

[15]

B. M. Miller and E. Y. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7.  Google Scholar

[16]

B. Øksendal and A. Sulem, Optimal consumption and portfolio with both fixed and proportional transaction costs, SIAM J. Control Optim., 40 (2002), 1765-1790. doi: 10.1137/S0363012900376013.  Google Scholar

[17]

L. Pan and J. Yong, A differential game with multi-level of hierarchy, J. Math. Anal. Appl., 161 (1991), 522-544. doi: 10.1016/0022-247X(91)90348-4.  Google Scholar

[18]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[19]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979. doi: 10.1137/0328054.  Google Scholar

[20]

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

[21]

G. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information, IEEE Trans. Autom. Control, 54 (2009), 1230-1242. doi: 10.1109/TAC.2009.2019794.  Google Scholar

[22]

G. Wang and Z. Yu, A Pontryagin's maximum principle for nonzero-sum differential games of BSDEs with applications, IEEE Trans. Autom. Control, 55 (2010), 1742-1747. doi: 10.1109/TAC.2010.2048052.  Google Scholar

[23]

G. Wang and Z. Yu, A partial information nonzero-sum differential game of backward stochastic diffrential equations with applications, Automatica, 48 (2012), 342-352. doi: 10.1016/j.automatica.2011.11.010.  Google Scholar

[24]

Z. Wu, Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems, Syst. Sci. Math. Sci., 11 (1998), 249-259.  Google Scholar

[25]

W. Xu, Stochastic maximum principle for optimal control problem of forward and backward system, Journal of the Australian Math. Society B, 37 (1995), 172-185. doi: 10.1017/S0334270000007645.  Google Scholar

[26]

D. W. K. Yeung and L. A. Petrosyan, Cooperative Stochastic Differential Games, Springer Series in Operations Research and Financial Engineering. Springer, New York, 2006.  Google Scholar

[27]

J. Yong, A leader-follower stochastic linear quadratic differential game, SIAM J. Control Optim., 41 (2002), 1015-1041. doi: 10.1137/S0363012901391925.  Google Scholar

show all references

References:
[1]

T. T. K. An and B. Øksendal, Maximum principle 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]

T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, Mathematics in Science and Engineering, 160. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.  Google Scholar

[3]

A. Bensoussan, Lectures on Stochastic Control, in Nonlinear Filtering and Stochastic Control, ser. Lecture Notes in Mathematics, New York: Springer Verlag, 1982.  Google Scholar

[4]

A. Cadenillas and I. Karatzas, The stochastic maximum principle for linear convex optimal control with random coefficients, SIAM J. Control Optim., 33 (1995), 590-624. doi: 10.1137/S0363012992240722.  Google Scholar

[5]

A. Cadenillas and F. Zapatero, Classical and impulse stochastic control of the exchange rate using interest rates and reserves, Math. Finance, 10 (2000), 141-156. doi: 10.1111/1467-9965.00086.  Google Scholar

[6]

M. H. A. Davis and A. Norman, Portfolio selection with transaction costs, Math. Oper. Res., 15 (1990), 676-713. doi: 10.1287/moor.15.4.676.  Google Scholar

[7]

D. Duffie and L. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394. doi: 10.2307/2951600.  Google Scholar

[8]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.  Google Scholar

[9]

S. Hamadéne, Nonzero-sum linear-quadratic stochastic differential games and backward-forward equations, Stochastic Anal. Appl., 17 (1999), 117-130. doi: 10.1080/07362999908809591.  Google Scholar

[10]

E. C. M., 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]

R. Isaacs, Differential Games,, Parts 1-4. The RAND Corporation, (): 1.   Google Scholar

[12]

M. Jeanblanc-Pique, Impulse control method and exchange rate, Math. Finance, 3 (1993), 161-177. doi: 10.1111/j.1467-9965.1993.tb00085.x.  Google Scholar

[13]

R. Korn, Some appliations of impulse control in mathematical finance, Math. Meth. Oper. Res., 50 (1999), 493-518. doi: 10.1007/s001860050083.  Google Scholar

[14]

A. E. B. Lim and X. Zhou, Risk-sensitive control with HARA utility, IEEE Trans. Autom. Control, 46 (2001), 563-578. doi: 10.1109/9.917658.  Google Scholar

[15]

B. M. Miller and E. Y. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7.  Google Scholar

[16]

B. Øksendal and A. Sulem, Optimal consumption and portfolio with both fixed and proportional transaction costs, SIAM J. Control Optim., 40 (2002), 1765-1790. doi: 10.1137/S0363012900376013.  Google Scholar

[17]

L. Pan and J. Yong, A differential game with multi-level of hierarchy, J. Math. Anal. Appl., 161 (1991), 522-544. doi: 10.1016/0022-247X(91)90348-4.  Google Scholar

[18]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[19]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979. doi: 10.1137/0328054.  Google Scholar

[20]

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

[21]

G. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information, IEEE Trans. Autom. Control, 54 (2009), 1230-1242. doi: 10.1109/TAC.2009.2019794.  Google Scholar

[22]

G. Wang and Z. Yu, A Pontryagin's maximum principle for nonzero-sum differential games of BSDEs with applications, IEEE Trans. Autom. Control, 55 (2010), 1742-1747. doi: 10.1109/TAC.2010.2048052.  Google Scholar

[23]

G. Wang and Z. Yu, A partial information nonzero-sum differential game of backward stochastic diffrential equations with applications, Automatica, 48 (2012), 342-352. doi: 10.1016/j.automatica.2011.11.010.  Google Scholar

[24]

Z. Wu, Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems, Syst. Sci. Math. Sci., 11 (1998), 249-259.  Google Scholar

[25]

W. Xu, Stochastic maximum principle for optimal control problem of forward and backward system, Journal of the Australian Math. Society B, 37 (1995), 172-185. doi: 10.1017/S0334270000007645.  Google Scholar

[26]

D. W. K. Yeung and L. A. Petrosyan, Cooperative Stochastic Differential Games, Springer Series in Operations Research and Financial Engineering. Springer, New York, 2006.  Google Scholar

[27]

J. Yong, A leader-follower stochastic linear quadratic differential game, SIAM J. Control Optim., 41 (2002), 1015-1041. doi: 10.1137/S0363012901391925.  Google Scholar

[1]

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
[2]

Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115
[3]

Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control & Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010
[4]

Qingmeng Wei, Zhiyong Yu. Time-inconsistent recursive zero-sum stochastic differential games. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1051-1079. doi: 10.3934/mcrf.2018045
[5]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035
[6]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems & Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002
[7]

Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447
[8]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613
[9]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047
[10]

Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7
[11]

Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028
[12]

Ying Liu, Yabing Sun, Weidong Zhao. Explicit multistep stochastic characteristic approximation methods for forward backward stochastic differential equations. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021044
[13]

Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153
[14]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001
[15]

Mrinal K. Ghosh, Somnath Pradhan. A nonzero-sum risk-sensitive stochastic differential game in the orthant. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021025
[16]

Beatris A. Escobedo-Trujillo. Discount-sensitive equilibria in zero-sum stochastic differential games. Journal of Dynamics & Games, 2016, 3 (1) : 25-50. doi: 10.3934/jdg.2016002
[17]

Kai Du, Jianhui Huang, Zhen Wu. Linear quadratic mean-field-game of backward stochastic differential systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 653-678. doi: 10.3934/mcrf.2018028
[18]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075
[19]

Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803
[20]

Joscha Diehl, Jianfeng Zhang. Backward stochastic differential equations with Young drift. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 5-. doi: 10.1186/s41546-017-0016-5

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (104)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences