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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

 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.
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.

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##### 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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