Optimal control of mean-field backward doubly stochastic systems driven by Itô-Lévy processes

References

• Andersson, D., & Djehiche, B. (2011). A maximum principle for SDEs of mean-field type. Applied Mathematics and Optimization, 63, 341–356. doi: 10.1007/s00245-010-9123-8
   Web of Science ®Google Scholar
• Applebaum, D. (2009). Lévy processes and stochastic calculus (2nd ed.). Cambridge, UK: Cambridge University Press.
   Google Scholar
• Bahlali, S., & Gherbal, B. (2010). Optimality conditions of controlled backward doubly stochastic differential equations. Random Operators and Stochastic Equations, 18, 247–265.
   Google Scholar
• Bally, V., & Matoussi, A. (2001). Weak solutions for SPDEs and backward doubly stochastic differential equations. Journal of Theoretical Probability, 14, 125–164. doi: 10.1023/A:1007825232513
   Web of Science ®Google Scholar
• Buckdahn, R., Djehiche, B., & Li, J. (2011). A general stochastic maximum principle for SDEs of mean-field type. Applied Mathematics and Optimization, 64, 197–216. doi: 10.1007/s00245-011-9136-y
   Web of Science ®Google Scholar
• Buckdahn, R., Djehiche, B., Li, J., & Peng, S. (2009). Mean-field backward stochastic differential equations: A limit approach. The Annals of Probability, 37, 1524–1565. doi: 10.1214/08-AOP442
   Web of Science ®Google Scholar
• Buckdahn, R., Li, J., Peng, S., & Rainer, C. (2017). Mean-field stochastic differential equations and associated PDEs. The Annals of Probability, 45, 824–878. doi: 10.1214/15-AOP1076
   Web of Science ®Google Scholar
• Dahl, K., Mohammed, S.-E., Øksendal, B., & Røse, E. (2016). Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives. Journal of Functional Analysis, 271, 289–329. doi: 10.1016/j.jfa.2016.04.031
   Web of Science ®Google Scholar
• Han, Y., Peng, S., & Wu, Z. (2010). Maximum principle for backward doubly stochastic control systems with applications. SIAM Journal on Control and Optimization, 48, 4224–4241. doi: 10.1137/080743561
   Web of Science ®Google Scholar
• Jacod, J., & Protter, P. (1988). Time reversal on Lévy processes. The Annals of Probability, 16, 620–641. doi: 10.1214/aop/1176991776
   Web of Science ®Google Scholar
• Li, J. (2012). Stochastic maximum principle in the mean-field controls. Automatica, 48, 366–373. doi: 10.1016/j.automatica.2011.11.006
   Web of Science ®Google Scholar
• Løkka, A. (2004). Martingale representation of functionals of Lévy processes. Stochastic Analysis and Applications, 22, 867–892. doi: 10.1081/SAP-120037622
   Web of Science ®Google Scholar
• Matoussi, A., Sabbagh, W., & Zhang, T. (2017). Backward doubly SDEs and semilinear stochastic PDEs in a convex domain. Stochastic Processes and their Applications, 127, 2781–2815. doi: 10.1016/j.spa.2016.12.010
   Web of Science ®Google Scholar
• Nualart, D., & Pardoux, E. (1988). Stochastic calculus with anticipating integrands. Probability Theory and Related Fields, 78, 535–581. doi: 10.1007/BF00353876
   Web of Science ®Google Scholar
• Øksendal, B., & Sulem, A. (2009). Applied stochastic control of jump diffusions. Berlin: Springer.
   Google Scholar
• Pardoux, E., & Peng, S. (1994). Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probability Theory and Related Fields, 98, 209–227. doi: 10.1007/BF01192514
   Web of Science ®Google Scholar
• Peng, S., & Shi, Y. (2003). A type of time-symmetric forward-backward stochastic differential equations. Comptes Rendus Mathematique, 336, 773–778. doi: 10.1016/S1631-073X(03)00183-3
   Web of Science ®Google Scholar
• Prato, G., Menaldi, J., & Tubaro, L. (2007). Some results of backward Itô formula. Stochastic Analysis and Applications, 25, 679–703. doi: 10.1080/07362990701283045
   Web of Science ®Google Scholar
• Ren, Y., Lin, A., & Hu, L. (2009). Stochastic PDIEs and backward doubly stochastic differential equations driven by Lévy processes. Journal of Computational and Applied Mathematics, 223, 901–907. doi: 10.1016/j.cam.2008.03.008
   Web of Science ®Google Scholar
• Shen, Y., & Siu, T. (2013). The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem. Nonlinear Analysis. Theory, Methods & Applications, 86, 58–73. doi: 10.1016/j.na.2013.02.029
   Web of Science ®Google Scholar
• Shi, Y., Gu, Y., & Liu, K. (2005). Comparison theorems of backward doubly stochastic differential equations and applications. Stochastic Analysis and Applications, 23, 97–110. doi: 10.1081/SAP-200044444
   Web of Science ®Google Scholar
• Shi, Y., & Zhu, Q. (2013). Partially observed optimal controls of forward-backward doubly stochastic systems. ESAIM: Control, Optimisation and Calculus of Variations, 19, 828–843. doi: 10.1051/cocv/2012035
   Web of Science ®Google Scholar
• Wang, G., Zhang, C., & Zhang, W. (2014). Stochastic maximum principle for mean-field type optimal control under partial information. IEEE Transactions on Automatic Control, 59, 522–528. doi: 10.1109/TAC.2013.2273265
   Web of Science ®Google Scholar
• Wu, J., & Liu, Z. (2017). Maximum principle for mean-field zero-sum stochastic differential game with partial information and its application to finance. European Journal of Control, 37, 8–15. doi: 10.1016/j.ejcon.2017.04.006
   Web of Science ®Google Scholar
• Wu, Z., & Zhang, F. (2011). BDSDEs with locally monotone coefficients and sobolev solutions for SPDEs. Journal of Differential Equations, 251, 759–784. doi: 10.1016/j.jde.2011.05.017
   Web of Science ®Google Scholar
• Xu, R. (2012). Mean-field backward doubly stochastic differential equations and related SPDEs. Boundary Value Problems, 2012, 114. doi: 10.1186/1687-2770-2012-114
   Web of Science ®Google Scholar
• Yong, J., & Zhou, X. (1999). Stochastic controls: Hamiltonian systems and HJB equations. New York, NY: Springer.
   Google Scholar
• Zhang, L., & Shi, Y. (2011). Maximum principle for forward-backward doubly stochastic control systems and applications. ESAIM: Control, Optimisation and Calculus of Variations, 17, 1174–1197. doi: 10.1051/cocv/2010042
   Web of Science ®Google Scholar
• Zhang, Q., & Zhao, H. (2007). Stationary solutions of SPDEs and infinite horizon BDSDEs. Journal of Functional Analysis, 252, 171–219. doi: 10.1016/j.jfa.2007.06.019
   Web of Science ®Google Scholar
• Zhang, Q., & Zhao, H. (2010). Stationary solutions of SPDEs and infinite horizon BDSDEs with non-lipschitz coefficients. Journal of Differential Equations, 248, 953–991. doi: 10.1016/j.jde.2009.12.013
   Web of Science ®Google Scholar
• Zhu, Q., & Shi, Y. (2015). Optimal control of backward doubly stochastic systems with partial information. IEEE Transactions on Automatic Control, 60, 173–178. doi: 10.1109/TAC.2014.2322212
   Web of Science ®Google Scholar
• Zhu, Q., Wang, X., & Shi, Y. (2013). Maximum principles for backward doubly stochastic systems with jumps and applications (in chinese). SCIENCE CHINA Mathematics, 43, 1237–1257.
   Google Scholar