# American Institute of Mathematical Sciences

September & December  2018, 8(3&4): 653-678. doi: 10.3934/mcrf.2018028

## Linear quadratic mean-field-game of backward stochastic differential systems

 1 School of Mathematics, Shandong University, Jinan 250100, China 2 Zhongtai Securities Institute for Financial Study, Shandong University, Jinan 250100, China 3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

* Corresponding author: Zhen Wu

Received  March 2017 Revised  December 2017 Published  September 2018

Fund Project: The work of K. Du is partially supported by the PolyU-SDU Joint Research Center (JRC) on Financial Mathematics. K. Du acknowledges the National Natural Sciences Foundations of China (11601285). J. Huang acknowledges the financial support by RGC Grant PolyU 153005/14P, 153275/16P, 5006/13P. Z. Wu acknowledges the Natural Science Foundation of China (61573217), the National High-level personnel of special support program and the Chang Jiang Scholar Program of Chinese Education Ministry.

This paper is concerned with a dynamic game of N weakly-coupled linear backward stochastic differential equation (BSDE) systems involving mean-field interactions. The backward mean-field game (MFG) is introduced to establish the backward decentralized strategies. To this end, we introduce the notations of Hamiltonian-type consistency condition (HCC) and Riccati-type consistency condition (RCC) in BSDE setup. Then, the backward MFG strategies are derived based on HCC and RCC respectively. Under mild conditions, these two MFG solutions are shown to be equivalent. Next, the approximate Nash equilibrium of derived MFG strategies are also proved. In addition, the scalar-valued case of backward MFG is solved explicitly. As an illustration, one example from quadratic hedging with relative performance is further studied.

Citation: 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
##### References:
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Wu, Adapted solution of generalized forward-backward stochastic differential equations and its dependence on parameters, Chinese Journal of Contemporary Mathematics, 19 (1998), 9-18.   Google Scholar [39] J. Yong, A linear-quadratic optimal control problem for mean-field stochastic differential equations, SIAM Journal on Control and Optimization, 51 (2013), 2809-2838.  doi: 10.1137/120892477.  Google Scholar [40] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar [41] D. T. Zhang, Forward-backward stochastic differential equations and backward linear quadratic stochastic optimal control problem, Communications in Mathematical Reserach, 25 (2009), 402-410.   Google Scholar

show all references

##### References:
 [1] A. B. Abel, Asset prices under habit formation and catching up with the Joneses, The American Economic Review, 80 (1990), 38-42.   Google Scholar [2] D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Applied Mathematics and Optimization, 63 (2011), 341-356.  doi: 10.1007/s00245-010-9123-8.  Google Scholar [3] M. Bardi, Explicit solutions of some linear-quadratic mean field games, Networks and Heterogeneous Media, 7 (2012), 243-261.  doi: 10.3934/nhm.2012.7.243.  Google Scholar [4] A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and mean Field Type Control Theory, Springerbriefs in Mathematics, 2013. doi: 10.1007/978-1-4614-8508-7.  Google Scholar [5] A. Bensoussan, K. Sung, S. Yam and S. Yung, Linear-quadratic mean-field games, Journal of Optimization Theory and Applications, 169 (2016), 496-529.  doi: 10.1007/s10957-015-0819-4.  Google Scholar [6] J. Bismut, An introductory approach to duality in optimal stochastic control, SIAM Review, 20 (1978), 62-78.  doi: 10.1137/1020004.  Google Scholar [7] R. Buckdahn, B. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Applied Mathematics and Optimization, 64 (2011), 197-216.  doi: 10.1007/s00245-011-9136-y.  Google Scholar [8] R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM Journal on Control and Optimization, 51 (2013), 2705-2734.  doi: 10.1137/120883499.  Google Scholar [9] Y. L. Chan and L. Kogan, Catching up with the Joneses: Heterogeneous preferences and the dynamics of asset prices, Journal of Political Economy, 110 (2002), 1255-1185.   Google Scholar [10] P. DeMarzo, R. Kaniel and I. Kremer, Relative wealth concerns and financial bubbles, Review of Financial Studies, 21 (2008), 19-50.   Google Scholar [11] D. Duffie, Dynamic Asset Pricing Theory, 3rd Edition, Princeton University Press, 2010. Google Scholar [12] D. Duffie and H. R. Richardson, Mean-variance hedging in continuous time, The Annals of Applied Probability, 1 (1991), 1-15.  doi: 10.1214/aoap/1177005978.  Google Scholar [13] R. Elliott and T. Siu, A BSDE approach to a risk-based optimal investment of an insurer, Automatica, 47 (2011), 253-261.  doi: 10.1016/j.automatica.2010.10.032.  Google Scholar [14] G. E. Espinosa and N. Touzi, Optimal investment under relative performance concerns, Mathematical Finance, 25 (2015), 221-257.  doi: 10.1111/mafi.12034.  Google Scholar [15] N. El Karoui, S. Peng and M. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.  Google Scholar [16] G. Guan and Z. Liang, Optimal management of DC pension plan under loss aversion and value-at-risk constraints, Insurance: Mathematics and Ecnomics, 69 (2016), 224-237.  doi: 10.1016/j.insmatheco.2016.05.014.  Google Scholar [17] J. Huang, S. Wang and Z. Wu, Backward mean-field linear-quadratic-gaussian (LQG) games: Full and partial information, IEEE Transactions on Automatic Control, 61 (2016), 3784-3796.  doi: 10.1109/TAC.2016.2519501.  Google Scholar [18] M. Huang, Large-population LQG games involving a major player: The Nash certainty equivalence principle, SIAM Journal on Control and Optimization, 48 (2010), 3318-3353.  doi: 10.1137/080735370.  Google Scholar [19] M. Huang, P. Caines and R. Malhamé, Large-population cost-coupled LQG problems with non-uniform agents: Individual-mass behavior and decentralized $\varepsilon$-Nash equilibria, IEEE Transactions on Automatic Control, 52 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450.  Google Scholar [20] M. Huang, R. Malhamé and P. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Communication in Information and Systems, 6 (2006), 221-251.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar [21] M. Kohlmann and X. Y. Zhou, Relationship between backward stochastic differential equations and stochsdtic controls: A linear-quadratic approach, SIAM Journal on Control and Optimization, 38 (2000), 1392-1407.  doi: 10.1137/S036301299834973X.  Google Scholar [22] J. M. Lasry and P. L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.  Google Scholar [23] T. Li and J. Zhang, Asymptotically optimal decentralized control for large population stochastic multiagent systems, IEEE Transactions on Automatic Control, 53 (2008), 1643-1660.  doi: 10.1109/TAC.2008.929370.  Google Scholar [24] A. E. Lim, Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Mathematics of Operations Research, 29 (2004), 132-161.  doi: 10.1287/moor.1030.0065.  Google Scholar [25] A. E. Lim and X. Y. Zhou, Linear-quadratic control of backward stochastic differential equations, SIAM Journal on Control and Optimization, 40 (2001), 450-474.  doi: 10.1137/S0363012900374737.  Google Scholar [26] M. Loève, Probability Theory, 4th Edition, New York: Springer-Verlag, 1978.  Google Scholar [27] J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and Their Applications, Springer-Verlag, Berlin Heidelberg, 1999.  Google Scholar [28] D. Majerek, W. Nowak and W. Zieba, Conditional strong law of large number, International Journal of Pure and Applied Mathematics, 20 (2005), 143-157.   Google Scholar [29] S. Nguyen and M. Huang, Linear-quadratic-Gaussian mixed games with continuum-parametrized minor players, SIAM Journal on Control and Optimization, 50 (2012), 2907-2937.  doi: 10.1137/110841217.  Google Scholar [30] N. Nolde and G. Parker, Stochastic analysis of life insurance surplus, Insurance: Mathematics and Economics, 56 (2014), 1-13.  doi: 10.1016/j.insmatheco.2014.02.006.  Google Scholar [31] M. Nourian and P. Caines, $\epsilon$-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents, SIAM Journal on Control and Optimization, 51 (2013), 3302-3331.  doi: 10.1137/120889496.  Google Scholar [32] E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems and Control Letters, 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.  Google Scholar [33] S. Peng, Backward stochastic differential equations and applications to optimal control, Applied Mathematics and Optimization, 27 (1993), 125-144.  doi: 10.1007/BF01195978.  Google Scholar [34] S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM Journal on Control and Optimization, 37 (1999), 825-843.  doi: 10.1137/S0363012996313549.  Google Scholar [35] D. Pirjol and L. Zhu, Discrete sums of geometric Brownian motions, annuities and asian options, Insurance: Mathematics and Economics, 70 (2016), 19-37.  doi: 10.1016/j.insmatheco.2016.05.020.  Google Scholar [36] H. Tembine, Q. Zhu and T. Basar, Risk-sensitive mean-field stochastic differential games, IEEE Trans. Automat. Control, 59 (2014), 835-850.  doi: 10.1109/TAC.2013.2289711.  Google Scholar [37] B. Wang and J. Zhang, Mean field games for large-population multiagent systems with markov jump parameters, SIAM Journal on Control and Optimization, 50 (2012), 2308-2334.  doi: 10.1137/100800324.  Google Scholar [38] Z. Wu, Adapted solution of generalized forward-backward stochastic differential equations and its dependence on parameters, Chinese Journal of Contemporary Mathematics, 19 (1998), 9-18.   Google Scholar [39] J. Yong, A linear-quadratic optimal control problem for mean-field stochastic differential equations, SIAM Journal on Control and Optimization, 51 (2013), 2809-2838.  doi: 10.1137/120892477.  Google Scholar [40] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar [41] D. T. Zhang, Forward-backward stochastic differential equations and backward linear quadratic stochastic optimal control problem, Communications in Mathematical Reserach, 25 (2009), 402-410.   Google Scholar
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