This paper obtains a maximum principle for switching diffusions with mean-field interactions. The motivation stems from a wide range of applications for networked control systems in which large-scale systems are encountered and mean-field interactions are involved. Because of the complexity due to the switching, little has been done for the associate control problems with mean-field interactions. The main ingredient of this work is the use of conditional mean-fields, which is distinct from the existing literature. Using the maximum principle, optimal controls of linear quadratic Gaussian controls with mean-field interactions for switching diffusions are carried out. Numerical examples are also provided for demonstration.
Accepté le :
Première publication :
Publié le :
DOI : 10.1051/cocv/2019055
Keywords: Maximum principle, mean-field interaction, switching diffusion
@article{COCV_2020__26_1_A69_0,
author = {Nguyen, Son L. and Nguyen, Dung T. and Yin, George},
title = {A stochastic maximum principle for switching diffusions using conditional mean-fields with applications to control problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2019055},
mrnumber = {4151428},
zbl = {1460.60082},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2019055/}
}
TY - JOUR AU - Nguyen, Son L. AU - Nguyen, Dung T. AU - Yin, George TI - A stochastic maximum principle for switching diffusions using conditional mean-fields with applications to control problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2019055/ DO - 10.1051/cocv/2019055 LA - en ID - COCV_2020__26_1_A69_0 ER -
%0 Journal Article %A Nguyen, Son L. %A Nguyen, Dung T. %A Yin, George %T A stochastic maximum principle for switching diffusions using conditional mean-fields with applications to control problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2019055/ %R 10.1051/cocv/2019055 %G en %F COCV_2020__26_1_A69_0
Nguyen, Son L.; Nguyen, Dung T.; Yin, George. A stochastic maximum principle for switching diffusions using conditional mean-fields with applications to control problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 69. doi: 10.1051/cocv/2019055
[1] and , A maximum principle for SDEs of mean-field type. Appl. Math. Optim. 63 (2011) 341–356. | MR | Zbl | DOI
[2] , , and , Real options with competition and regime switching. Math. Finance 27 (2017) 224–250. | MR | Zbl | DOI
[3] and , The Vlasov dynamics and its fluctuations in the 1/n limit of interacting classical particles. Commun. Math. Phys. 56 (1977) 101–113. | MR | Zbl | DOI
[4] , and , A mean-field stochastic control problem with partial observations. Ann. Appl. Probab. 27 (2017) 3201–3245. | MR | Zbl | DOI
[5] and , A probabilistic approach to mean field games with major and minor players. Ann. Appl. Probab. 26 (2016) 1535–1580. | MR | Zbl | DOI
[6] , Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Stat. Phys. 31 (1983) 29–85. | MR | DOI
[7] and , Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20 (1987) 247–308. | MR | Zbl | DOI
[8] and , A Hilbertian approach for fluctuations on the McKean-Vlasov model. Stoch. Process Appl. 71 (1997) 33–53. | MR | Zbl | DOI
[9] , On the McKean-Vlasov limit for interacting diffusions. Math. Nachr. 137 (1988) 197–248. | MR | Zbl | DOI
[10] , and , Tracking a Markov-modulated stationary degree distribution of a dynamic random graph, IEEE Trans. Inf. Theory 60 (2014) 6609–6625. | MR | Zbl | DOI
[11] and , Mean field games for stochastic growth with relative utility. Appl. Math. Optim. 74 (2016) 643–668. | MR | Zbl | DOI
[12] , and , Individual and mass behavior in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions. Proc. of 42nd IEEE CDC (2003) 98–103.
[13] , and , Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6 (2006) 221–252. | MR | Zbl | DOI
[14] , and , Social optima in mean field LQG control: centralized and decentralized strategies. IEEE Trans. Automat. Control 57 (2012) 1736–1751. | MR | Zbl | DOI
[15] and , Limit Theorems for Stochastic Processes, 2nd edn. Springer Science & Business Media, Berlin (2013). | Zbl
[16] and , On the mean field games with common noise and the McKean-Vlasov SPDEs. Preprint (2015). | arXiv | MR
[17] and , Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343 (2006) 619–625. | MR | Zbl | DOI
[18] , Stochastic maximum principle in the mean-field controls. Automatica 48 (2012) 366–373. | MR | Zbl | DOI
[19] and , Weak necessary and sufficient stochastic maximum principle for Markovian regime-switching diffusion models. Appl. Math. Optim. 71 (2015) 39–77. | MR | Zbl | DOI
[20] and , Theory of Martingales. Kluwer Academic Publishers Group, Dordrecht (1989) | MR | Zbl | DOI
[21] and , Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006). | MR | Zbl | DOI
[22] , A class of Markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. U.S.A. 56 (1966) 1907–1911. | MR | Zbl | DOI
[23] and , Linear-quadratic-Gaussian mixed games with continuum-parametrized minor player. SIAM J. Control Optim. 50 (2012) 2907–2937. | MR | Zbl | DOI
[24] , and , On mean field systems with multi-classes. Discrete Contin. Dyn. Syst. 40 (2020) 683–707. | MR | Zbl | DOI
[25] , and , Laws of large numbers for systems with mean-field interactions and Markovian switching. Stoch. Process Appl. 130 (2020) 262–296. | MR | Zbl | DOI
[26] , A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28 (1990) 966–979. | MR | Zbl | DOI
[27] and , Dynamic programming for optimal control of stochastic McKean–Vlasov dynamics. SIAM J. Control Optim. 55 (2017) 1069–1101. | MR | Zbl | DOI
[28] , Stochastic Integration and Differential Equations, 2nd edn., Springer, Berlin (2005) | MR | Zbl | DOI
[29] and , Markov Processes, and Martingales. Vol. 2. Itô Calculus (Reprint of the second (1994) edition). Cambridge University Press, Cambridge (2000). | MR | Zbl
[30] and , Nonlinear filtering theory for McKean-Vlasov type stochastic differential equations. SIAM J. Control Optim. 54 (2016), 153–174. | MR | Zbl | DOI
[31] , and , Leader-follower stochastic differential game with asymmetric information and applications. Automatica 63 (2016) 60–73. | MR | Zbl | DOI
[32] , Topics in propagation of chaos, in Ecole d?Eté de Probabilités de Saint-Flour XIX - 1989, edited by , Vol 1464 of Lecture Notes in Math. Springer-Verlag, Berlin (1991) 165–251. | MR | Zbl
[33] , Limit theorems for certain diffusion processes with interaction. Stoch. Anal. 32 (1984) 469–488. | MR | Zbl
[34] and , Mean field production output control with sticky prices: Nash and social solutions. Automatica 100 (2019) 90–98. | MR | Zbl | DOI
[35] and , Mean field games for large-population multiagent systems with Markov jump parameters. SIAM J. Control Optim. 50 (2012) 2308–2334. | MR | Zbl | DOI
[36] and , Social optima in mean field linear-quadratic-Gaussian models with Markov jump parameters. SIAM J. Control Optim. 55 (2017) 429–456. | MR | DOI
[37] and , Asymptotic properties of a mean-field model with a continuous-state-dependent switching process. J. Appl. Probab. 46 (2009) 221–243. | MR | Zbl | DOI
[38] and , Hybrid Switching Diffusions: Properties and Applications. Springer, New York (2010). | MR | Zbl | DOI
[39] , and , Regime switching stochastic approximation algorithms with application to adaptive discrete stochastic optimization. SIAM J. Optim. 14 (2004) 1187–1215. | MR | Zbl | DOI
[40] , and , Stochastic recursive algorithms for networked systems with delay and random switching: Multiscale formulations and asymptotic properties. SIAM J.: Multiscale Model. Simul. 9 (2011) 1087–1112. | MR | Zbl
[41] and , Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999). | MR | Zbl | DOI
[42] and , On nearly optimal controls of hybrid LQG problems. IEEE Trans. Automat. Control 44 (1999) 2271–2282. | MR | Zbl | DOI
[43] , and , A general stochastic maximum principle for a Markov regime switching jump-diffusion model of mean-field type. SIAM J. Control Optim. 56 (2018) 2563–2592. | MR | Zbl | DOI
Cité par Sources :
The research of S. Nguyen was supported by a seed fund of Department of Mathematics at University of Puerto Rico, Rio Piedras campus; the research of D. Nguyen was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03-2015.28; the research of G. Yin was supported in part by the Army Research Office under grant W911NF-19-1-0176.
