Stochastic optimal control for dynamics of forward backward doubly SDEs of mean-field type
DOI:
https://doi.org/10.5269/bspm.50597Abstract
In this paper we establish in first the existence of strong optimal solutions of a control problem for dynamics driven by a linear forward-backward doubly stochastic differential equations of mean-field type (MF-FBDSDEs), with random coefficients and non linear functional cost. Moreover, we establish necessary as well as sufficient optimality conditions for this kind of control problem. In the second part of this paper, we establish necessary as well as sufficient optimality conditions for existence of both optimal relaxed control and optimal strict control for dynamics of nonlinear mean-field forward-backward doubly stochastic differential equations.
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2. Andersson, D. and Djehiche, B., A maximum principle for SDEs of mean-field type, Appl. Math. and Optim, 63, no. 3, 341-356, (2010). https://doi.org/10.1007/s00245-010-9123-8
3. Al-Hussein, A. and Gherbal, B., Existence and uniqueness of the solutions of forward-backward doubly stochastic differential equations with Poisson jumps, arXiv:1809.01875 [math.PR], Submitted, (2018).
4. Benbrahim, R. and Gherbal, B., Existence of Optimal Controls for Forward-Backward Stochastic Differential Equations of Mean-Field Type, J. of Numerical Mathematics and Stochastics, 9, no. 1, 33-47, (2017).
5. Buckdahn, R., Djehiche, B. and Li, J., A general stochastic maximum principle for SDEs of mean-field type, Appl. Math. Optim. 64, no. 2, 197-216, (2011). https://doi.org/10.1007/s00245-011-9136-y
6. Buckdahn, R., Djehiche, B., Li, J. and Peng, S., Mean-Field backward stochastic differential equations: a limit approach, Ann. Prob., 37, no. 4, 1524-1565, (2009). https://doi.org/10.1214/08-AOP442
7. Carmona, R. and Delarue, F., Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51, no. 4, 2705-2734, (2013). https://doi.org/10.1137/120883499
8. Dawson, D., Critical dynamics and fluctuations for a mean-field model of cooperative behavoir. Journal of Statistical Physics, 31, no. 1, 29-85, (1983). https://doi.org/10.1007/BF01010922
9. Ekeland, I. and Temam, R., Analyse convexe et probl'eme variationnel, Dunod (1974).
10. Fu, G. and Horst, U., Mean Field Games with Singular Controls, SIAM J. Control and Optim., 55, no. 6, 3833-3868, (2017). https://doi.org/10.1137/17M1123742
11. Gherbal, B., Optimal control problems for linear backward doubly stochastic differential equations. Random. Oper. Stoc. Equ., 22, no. 3, 129-138, (2014). https://doi.org/10.1515/rose-2014-0014
12. Hafayed, M., A mean-field maximum principle for optimal control of forward-backward stochastic differential equations with Poisson jump processes, Int. J. Dyn. Control 1, no. 4, 300-315, (2013). https://doi.org/10.1007/s40435-013-0027-8
13. Huang, M. Y., Malhame, R. and Caines, P., Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Communications in information and systems, 6, no 3, 221-252, (2006). https://doi.org/10.4310/CIS.2006.v6.n3.a5
14. Li, R. and Liu, B., A maximum principle for fully coupled stochastic control systems of mean-field type, J. Math. Anal. Appl, 415, 902-930, (2014). https://doi.org/10.1016/j.jmaa.2014.02.008
15. Lasry, J.M. and Lions, P.L., Mean-field games. Japan. J. Math. 2, 229-260, (2007). https://doi.org/10.1007/s11537-007-0657-8
16. Meyer-Brandis, T. Øksendal, B. and Zhou, X.Y., A mean-field stochastic maximum principle via Malliavin calculus, Stochastics 84, 643-666, (2012). https://doi.org/10.1080/17442508.2011.651619
17. Wu, J. and Liu, Z., Optimal control of mean-field backward doubly stochastic systems driven by Itˆo-Levy processes. International Journal of control, (2018). https://doi.org/10.1080/00207179.2018.1502473
18. Xu, R. M., Mean field backward doubly stochastic differential equations and related SPDEs, Boundary Volue Problems, vol, article 114, (2012). https://doi.org/10.1186/1687-2770-2012-114
19. Zhu, Q. and Shi, Y., Mean-field Forward-Backward Doubly Stochastic Differential Equations and Related Nonlocal Stochastic Partial Differential Equations, Abstract and Applied Analysis, Hindawi, Article ID 194341, 10 pages, (2014). https://doi.org/10.1155/2014/194341
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2022-12-23
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Direction Générale de la Recherche Scientifique et du Développement Technologique
Grant numbers C00L03UN070120180005
