A mean-field optimal control for fully coupled forward-backward stochastic control systems with Lévy processes. (English) Zbl 1485.93631
Summary: This paper is concerned with a class of mean-field type stochastic optimal control systems, which are governed by fully coupled mean-field forward-backward stochastic differential equations with Teugels martingales associated to Lévy processes. In these systems, the coefficients contain not only the state processes but also their marginal distribution, and the cost function is of mean-field type as well. The necessary and sufficient conditions for such optimal problems are obtained. Furthermore, the applications to the linear quadratic stochastic optimization control problem are investigated.
MSC:
| 93E20 | Optimal stochastic control |
| 49N80 | Mean field games and control |
| 49N10 | Linear-quadratic optimal control problems |
| 60G51 | Processes with independent increments; Lévy processes |
Keywords:
adjoint equation; Lévy processes; mean-field forward-backward stochastic differential equations; stochastic maximum principle; Teugels martingalesReferences:
| [1] | Peng, S. G., A general maximum principle for optimal control problems, SIAM Journal on Control and Optimization, 28, 966-979 (1990) · Zbl 0712.93067 · doi:10.1137/0328054 |
| [2] | Peng, S. G., Backward stochastic differential equations and application to optimal control, Applied Mathematics and Optimization, 27, 2, 125-144 (1993) · Zbl 0769.60054 · doi:10.1007/BF01195978 |
| [3] | Wu, Z., Maximum principle for optimal control problem of fully coupled forward-backward stochastic control system, Journal of Systems Science and Mathematical Sciences, 11, 3, 249-259 (1998) · Zbl 0938.93066 |
| [4] | Shi, J.; Wu, Z., The maximum principle for fully coupled forward-backward stochastic control system, Acta Autom Sin, 32, 2, 161-169 (2006) · Zbl 1498.93786 |
| [5] | Shi, J.; Wu, Z., Maximum principle for fully coupled forward-backward stochastic control system with random jumps and applications to finance, Journal of Systems Science and Complexity, 23, 2, 219-231 (2010) · Zbl 1197.93165 · doi:10.1007/s11424-010-7224-8 |
| [6] | Tang, M. N.; Zhang, Q., Optimal variational principle for backward stochastic control systems associated with Lévy processes, Sci. China Mathematics, 55, 4, 745-761 (2012) · Zbl 1242.93148 · doi:10.1007/s11425-012-4370-6 |
| [7] | Zhang, F.; Tang, M. N.; Meng, Q. X., Sochastic maximum principle for forward-backward stochastic control systems associated with Lévy processes, Chinese Annals of Mathematics, 35A, 1, 83-100 (2014) · Zbl 1313.60116 · doi:10.1007/s11401-011-0690-z |
| [8] | Framstad, N.; Økesendal, B.; Sulem, A., Sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance, Journal of Optimization Theory and Applications, 121, 1, 77-98 (2004) · Zbl 1140.93496 · doi:10.1023/B:JOTA.0000026132.62934.96 |
| [9] | Lin, X.; Zhang, W., A maximum principle for optimal control of discrete-time stochastic systems with multiplicative noise, IEEE Transactions on Automatic Control, 60, 4, 1121-1126 (2015) · Zbl 1360.93395 · doi:10.1109/TAC.2014.2345243 |
| [10] | Zhang, X.; Elliott, R. J.; Siu, T., A stochastic maximum principle for a markov regime-switching jump-diffusion model and its application to finance, SIAM J. Control Optim., 50, 2, 964-990 (2012) · Zbl 1244.93180 · doi:10.1137/110839357 |
| [11] | Buckdahn, R.; Li, J.; Peng, S., Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Processes and Their Applications, 119, 10, 3133-3154 (2009) · Zbl 1183.60022 · doi:10.1016/j.spa.2009.05.002 |
| [12] | Wang, G.; Zhang, C.; Zhang, W., Stochastic maximum principle for mean-field type optimal control with partial information, IEEE Transactions on Automatic Control, 59, 2, 522-528 (2014) · Zbl 1360.93788 · doi:10.1109/TAC.2013.2273265 |
| [13] | Wang, G.; Xiao, H.; Xing, G., An optimal control problem for mean-field forward-backward stochastic differential equation with noisy observation, Automatica, 86, 104-109 (2017) · Zbl 1375.93143 · doi:10.1016/j.automatica.2017.07.018 |
| [14] | Agram, N.; Røse, E. E., Optimal control of forward-backward mean-field stochastic delayed systems, Afrika Matematika, 29, 149-174 (2018) · Zbl 1399.93235 · doi:10.1007/s13370-017-0532-6 |
| [15] | Wang, B. C.; Zhang, J. F., Mean field games for large-population multiagent systems with Markov jump parameters, SIAM Journal on Control and Optimization, 50, 4, 2308-2334 (2012) · Zbl 1263.91007 · doi:10.1137/100800324 |
| [16] | Kolokoltsov, V. N.; Malafeyev, O. A., Mean-field-game model of corruption, Dynamic Games & Applications, 7, 1, 34-47 (2017) · Zbl 1391.91030 · doi:10.1007/s13235-015-0175-x |
| [17] | Ma, H.; Liu, B., Optimal control problem for risk-sensitive mean-field stochastic delay differential equation with partial information, Asian Journal of Control, 19, 6, 2097-2115 (2017) · Zbl 1386.93311 · doi:10.1002/asjc.1570 |
| [18] | Wang, G.; Wang, Y.; Zhang, S., An asymmetric information mean-field type linear-quadratic stochastic Stackelberg differential game with one leader and two followers, Optimal Control Applications and Methods, 41, 4, 1034-1051 (2020) · Zbl 1469.91011 · doi:10.1002/oca.2585 |
| [19] | Nualart, D.; Schoutens, W., Chaotic and predictable representations for Lévy processes, Stochastic Processes and Their Applications, 90, 1, 109-122 (2000) · Zbl 1047.60088 · doi:10.1016/S0304-4149(00)00035-1 |
| [20] | Meng, Q. X.; Tang, M. N., Necessary and sufficient conditions for optimal control of stochastic systems associated with Lévy processes, Sci. China, Information Sciences, 52, 1982-1992 (2009) · Zbl 1182.49023 · doi:10.1007/s11432-009-0191-9 |
| [21] | Tang, H. B.; Wu, Z., Stochastic differential equations and stochastic linear quadratic optimal control problem with Lévy Process, Journal of Systems Science and Complexity, 22, 1, 122-136 (2009) · Zbl 1178.93148 · doi:10.1007/s11424-009-9151-0 |
| [22] | Wang X R and Huang H, Maximum principle for forward-backward stochastic control system driven by Lévy Process, Mathematical Problems in Engineering, 2015, ID 702802, 1-12. · Zbl 1394.49024 |
| [23] | Peng, S.; Wu, Z., Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM Journal on Control and Optimization, 37, 3, 825-843 (1999) · Zbl 0931.60048 · doi:10.1137/S0363012996313549 |
| [24] | Baghery, F.; Khelfallah, N.; Mezerdi, B., Fully coupled forward backward stochastic differential equations driven by Lévy processes and application to differential games, Random Operators & Stochastic Equations, 22, 3, 151-161 (2014) · Zbl 1300.60073 · doi:10.1515/rose-2014-0016 |
| [25] | Huang, M.; Caines, P. E.; Malhame, R. P., Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized ε-Nash equilibria, IEEE Transactions on Automatic Control, 52, 9, 1560-1571 (2007) · Zbl 1366.91016 · doi:10.1109/TAC.2007.904450 |
| [26] | Zhou, Q.; Ren, Y., A mean-field stochastic maximum principle for optimal control of forward-backward stochastic differential equations with jumps via Malliavin calculus, Journal of Applied Mathematics and Physics, 6, 1, 138-154 (2018) · doi:10.4236/jamp.2018.61014 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
