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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 and 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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

R. Isaacs, Differential Games, Parts 1-4. The RAND Corporation, Research Memorandums Nos. RM-1391, RM-1411,RM-1486, 1954-55.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[24]

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

[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.

[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.

[27]

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

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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

R. Isaacs, Differential Games, Parts 1-4. The RAND Corporation, Research Memorandums Nos. RM-1391, RM-1411,RM-1486, 1954-55.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[24]

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

[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.

[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.

[27]

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

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