×

Mean-field SDE driven by a fractional Brownian motion and related stochastic control problem. (English) Zbl 1361.93066

Summary: We study a class of mean-field stochastic differential equations driven by a fractional Brownian motion with Hurst parameter \(H\in(1/2,1)\) and a related stochastic control problem. We derive a Pontryagin-type maximum principle and the associated adjoint mean-field backward stochastic differential equation driven by a classical Brownian motion, and we prove that under certain assumptions, which generalize the classical ones. The necessary condition for the optimality of an admissible control is also sufficient.

MSC:

93E20 Optimal stochastic control
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65 Brownian motion
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
49K45 Optimality conditions for problems involving randomness
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] F. Biagini, Y. Hu, B. Øksendal, and A. Sulem, {\it A stochastic maximum principle for processes driven by fractional Brownian motion}, Stochastic Process. Appl., 100 (2002), pp. 233-253. · Zbl 1064.93048
[2] F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, {\it Stochastic Calculus for Fractional Brownian Motion and Applications}, Springer, New York, 2008. · Zbl 1157.60002
[3] R. Buckdahn, {\it Anticipative Girsanov transformations and Skorohod stochastic differential equations}, Mem. Amer. Math. Soc., 111 (1994). · Zbl 0849.60053
[4] R. Buckdahn, B. Djehiche, J. Li, and S. Peng, {\it Mean-field backward stochastic differential equations: A limit approach}, Ann. Probab., 37 (2009), pp. 1524-1565. · Zbl 1176.60042
[5] R. Buckdahn and S. Jing, {\it Peng’s maximum principle for a stochastic control problem driven by a fractional and a standard Brownian motion}, Sci. China Math., 57 (2014), pp. 2025-2042. · Zbl 1305.93205
[6] R. Buckdahn, J. Li, and S. Peng, {\it Mean-field backward stochastic differential equations and related partial differential equations}, Stochastic Process. Appl., 119 (2009), pp. 3133-3154. · Zbl 1183.60022
[7] P. Cardaliaguet, {\it Notes on Mean Field Games}, Technical report, (2010).
[8] R. Carmona and F. Delarue, {\it Forward€“ backward stochastic differential equations and controlled McKean Vlasov dynamics}, Ann. Probab., 43 (2015), pp. 2647-2700. · Zbl 1322.93103
[9] T. Duncan, Y. Hu, and B. Pasik-Duncan, {\it Stochastic calculus for fractional Brownian motion} I. {\it Theory}, SIAM J. Control Optim., 38 (2000), pp. 582-612. · Zbl 0947.60061
[10] Y. Han, Y. Hu, and J. Song, {\it Maximum principle for general controlled systems driven by fractional Brownian motions}, Appl. Math. Optim., 67 (2013), pp. 279-322. · Zbl 1270.49021
[11] Y. Hu, {\it Integral Transformations and Anticipative Calculus for Fractional Brownian Motions}, Mem. Amer. Math. Soc., 825 (2005). · Zbl 1072.60044
[12] Y. Hu and X. Zhou, {\it Stochastic control for linear systems driven by fractional noises}, SIAM J. Control Optim., 43 (2005), pp. 2245-2277. · Zbl 1116.93055
[13] S. Jing and J. A. León, {\it Semilinear backward doubly stochastic differential equations and SPDEs driven by fractional Brownian motion with Hurst parameter in} (0, 1/2), Bull. Sci. Math., 135 (2011), pp. 896-935. · Zbl 1242.60066
[14] M. Kac, {\it Foundations of kinetic theory}, in Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, Vol. 3, 1956, pp. 171-197. · Zbl 0072.42802
[15] M. Kac, {\it Probability and Related Topics in the Physical Sciences}, Interscience, New York, 1958.
[16] J. M. Lasry and P. L. Lions, {\it Mean field games}, Jpn. J. Math., 2 (2007), pp. 229-260. · Zbl 1156.91321
[17] Y. Mishura, {\it Stochastic Calculus for Fractional Brownian Motion and Related Processes}, Springer, New York, 2008. · Zbl 1138.60006
[18] D. Nualart, {\it The Malliavin Calculus and Related Topics}, 2nd ed., Springer-Verlag, Heidelberg, 2006. · Zbl 1099.60003
[19] S. Peng and Z. Wu, {\it Fully coupled forward-backward stochastic differential equations and applications to optimal control}, SIAM J. Control Optim., 37 (1999), pp. 825-843. · Zbl 0931.60048
[20] S. G. Samko, A. A. Kilbas, and O. I. Marichev, {\it Fractional Integrals and Derivatives: Theory and Applications}, Gordon and Breach, New York, 1987. · Zbl 0617.26004
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.