# American Institute of Mathematical Sciences

June  2017, 7(2): 289-304. doi: 10.3934/mcrf.2017010

## Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs

 1 Center for Financial Engineering, Soochow University, Suzhou 215006, China 2 School of Mathematics, Shandong University, Jinan 250100, China

* Corresponding author Zhen Wu.

Revised  January 2017 Published  April 2017

This paper is concerned with recursive nonzero-sum stochastic differential game problem in Markovian framework when the drift of the state process is no longer bounded but only satisfies the linear growth condition. The costs of players are given by the initial values of related backward stochastic differential equations which, in our case, are multidimensional with continuous coefficients, whose generators are of linear growth on the volatility processes and stochastic monotonic on the value processes. We finally show the well-posedness of the costs and the existence of a Nash equilibrium point for the game under the generalized Isaacs assumption.

Citation: Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control and Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010
##### References:
 [1] P. Briand and F. Confortola, BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Processes and their Applications, 118 (2008), 818-838.  doi: 10.1016/j.spa.2007.06.006. [2] R. Buckdahn, P. Cardaliaguet and M. Quincampoix, Some recent aspects of differential game theory, Dynamic Games and Applications, 1 (2011), 74-114.  doi: 10.1007/s13235-010-0005-0. [3] D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica: Journal of the Econometric Society, 60 (1992), 353-394.  doi: 10.2307/2951600. [4] N. El-Karoui and D. Hamadéne, BSDEs and risk-sensitive control, zero-sum and nonzero-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. [5] N. El-Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022. [6] I. V. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, Theory of Probability and its Applications, 5 (1960), 314-330. [7] S. Hamadéne, Backward-forward SDE's and stochastic differential games, Stochastic Processes and their Applications, 77 (1998), 1-15.  doi: 10.1016/S0304-4149(98)00038-6. [8] S. Hamadéne, J.-P. Lepeltier and S. Peng, BSDEs with continuous coefficients and stochastic differential games, Pitman Research Notes in Mathematics Series, 364 (1997), 115-128. [9] S. Hamadéne and R. Mu, Existence of Nash equilibrium points for Markovian non-zero-sum stochastic differential games with unbounded coefficients, Stochastics An International Journal of Probability and Stochastic Processes, 87 (2015), 85-111.  doi: 10.1080/17442508.2014.915973. [10] S. Hamadéne and R. Mu, Bang-bang-type Nash equilibrium point for Markovian nonzero-sum stochastic differential game, Comptes Rendus Mathematique, 352 (2014), 699-706.  doi: 10.1016/j.crma.2014.06.011. [11] S. Hamadéne and R. Mu, Risk-sensitive nonzero-sum stochastic differential game with unbounded coefficients, preprint, arXiv: 1412.1213. [12] U. G. Haussmann, A Stochastic Maximum Principle for Optimal Control of Diffusions, John Wiley & Sons, Inc. , 1986. [13] Q. Lin, A BSDE approach to Nash equilibrium payoffs for stochastic differential games with nonlinear cost functionals, Stochastic Processes and their Applications, 122 (2012), 357-385.  doi: 10.1016/j.spa.2011.08.011. [14] H. P. Mckean, Stochastic Integrals, Academic Press, New York-London, 1969. [15] M. A. Morlais, Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem, Finance and Stochastics, 13 (2009), 121-150.  doi: 10.1007/s00780-008-0079-3. [16] R. Mu and Z. Wu, One kind of multiple dimensional Markovian BSDEs with stochastic linear growth generators, Advances in Difference Equations, 2015 (2015), 1-15.  doi: 10.1186/s13662-015-0607-3. [17] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus -2nd ed. Springer Verlag, 1991. doi: 10.1007/978-1-4612-0949-2. [18] S. Peng, Backward stochastic differential equation, nonlinear expectation and their applications, Proceedings of the International Congress of Mathematicians, 1 (2010), 393-432. [19] G. Wang and Z. Wu, Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems, Journal of Mathematical Analysis and Applications, 342 (2008), 1280-1296.  doi: 10.1016/j.jmaa.2007.12.072. [20] L. Wei and Z. Wu, Stochastic recursive zero-sum differential game and mixed zero-sum differential game problem, Mathematical Problems in Engineering 2012 (2012), Art. ID 718714, 15 pp.

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##### References:
 [1] P. Briand and F. Confortola, BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Processes and their Applications, 118 (2008), 818-838.  doi: 10.1016/j.spa.2007.06.006. [2] R. Buckdahn, P. Cardaliaguet and M. Quincampoix, Some recent aspects of differential game theory, Dynamic Games and Applications, 1 (2011), 74-114.  doi: 10.1007/s13235-010-0005-0. [3] D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica: Journal of the Econometric Society, 60 (1992), 353-394.  doi: 10.2307/2951600. [4] N. El-Karoui and D. Hamadéne, BSDEs and risk-sensitive control, zero-sum and nonzero-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. [5] N. El-Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022. [6] I. V. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, Theory of Probability and its Applications, 5 (1960), 314-330. [7] S. Hamadéne, Backward-forward SDE's and stochastic differential games, Stochastic Processes and their Applications, 77 (1998), 1-15.  doi: 10.1016/S0304-4149(98)00038-6. [8] S. Hamadéne, J.-P. Lepeltier and S. Peng, BSDEs with continuous coefficients and stochastic differential games, Pitman Research Notes in Mathematics Series, 364 (1997), 115-128. [9] S. Hamadéne and R. Mu, Existence of Nash equilibrium points for Markovian non-zero-sum stochastic differential games with unbounded coefficients, Stochastics An International Journal of Probability and Stochastic Processes, 87 (2015), 85-111.  doi: 10.1080/17442508.2014.915973. [10] S. Hamadéne and R. Mu, Bang-bang-type Nash equilibrium point for Markovian nonzero-sum stochastic differential game, Comptes Rendus Mathematique, 352 (2014), 699-706.  doi: 10.1016/j.crma.2014.06.011. [11] S. Hamadéne and R. Mu, Risk-sensitive nonzero-sum stochastic differential game with unbounded coefficients, preprint, arXiv: 1412.1213. [12] U. G. Haussmann, A Stochastic Maximum Principle for Optimal Control of Diffusions, John Wiley & Sons, Inc. , 1986. [13] Q. Lin, A BSDE approach to Nash equilibrium payoffs for stochastic differential games with nonlinear cost functionals, Stochastic Processes and their Applications, 122 (2012), 357-385.  doi: 10.1016/j.spa.2011.08.011. [14] H. P. Mckean, Stochastic Integrals, Academic Press, New York-London, 1969. [15] M. A. Morlais, Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem, Finance and Stochastics, 13 (2009), 121-150.  doi: 10.1007/s00780-008-0079-3. [16] R. Mu and Z. Wu, One kind of multiple dimensional Markovian BSDEs with stochastic linear growth generators, Advances in Difference Equations, 2015 (2015), 1-15.  doi: 10.1186/s13662-015-0607-3. [17] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus -2nd ed. Springer Verlag, 1991. doi: 10.1007/978-1-4612-0949-2. [18] S. Peng, Backward stochastic differential equation, nonlinear expectation and their applications, Proceedings of the International Congress of Mathematicians, 1 (2010), 393-432. [19] G. Wang and Z. Wu, Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems, Journal of Mathematical Analysis and Applications, 342 (2008), 1280-1296.  doi: 10.1016/j.jmaa.2007.12.072. [20] L. Wei and Z. Wu, Stochastic recursive zero-sum differential game and mixed zero-sum differential game problem, Mathematical Problems in Engineering 2012 (2012), Art. ID 718714, 15 pp.
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