Abstract
We consider a multi-player stochastic differential game with linear McKean-Vlasov dynamics and quadratic cost functional depending on the variance and mean of the state and control actions of the players in open-loop form. Finite and infinite horizon problems with possibly some random coefficients as well as common noise are addressed. We propose a simple direct approach based on weak martingale optimality principle together with a fixed point argument in the space of controls for solving this game problem. The Nash equilibria are characterized in terms of systems of Riccati ordinary differential equations and linear mean-field backward stochastic differential equations: existence and uniqueness conditions are provided for such systems. Finally, we illustrate our results on a toy example.
This work is supported by FiME (Finance for Energy Market Research Centre) and the “Finance et Développement Durable - Approches Quantitatives” EDF - CACIB Chair.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Aurell and B. Djehiche. Mean-field type modeling of nonlocal crowd aversion in mean-field modeling of nonlocal crowd aversion in pedestrian crowd dynamics. SIAM Journal on Control and Optimization, 56(1):434–455, 2018.
M. Basei and H. Pham. A weak martingale approach to linear-quadratic mckean-vlasov stochastic control problems. Journal of Optimization Theory and Applications, to appear, 2018.
A. Bensoussan, J. Frehse, and P. Yam. Mean field games and mean field type control theory. Springer Briefs in Mathematics, 2013.
A. Bensoussan, T. Huang, and M. Laurière. Mean field control and mean field game models with several populations. arXiv:1810.00783.
R. Carmona and F. Delarue. Forward-backward stochastic differential equations and controlled mckean-vlasov dynamics. Annals of Probability, 43(5):2647–2700, 2015.
R. Carmona and F. Delarue. Probabilistic Theory of Mean Field Games with Applications vol I and II. Springer, 2018.
A. Cosso and H. Pham. Zero-sum stochastic differential games of generalized mckean-vlasov type. Journal de Mathématiques Pures et Appliquées, to appear, 2018.
B. Djehiche, J. Barreiro-Gomez, and H. Tembine. Electricity price dynamics in the smart grid: A mean-field-type game perspective. In 23rd International Symposium on Mathematical Theory of Networks and Systems (MTNS2018), pages 631–636, 2018.
T. Duncan and H. Tembine. Linear-quadratic mean-field-type games: a direct method. Games, 9(7), 2018.
P.J. Graber. Linear-quadratic mean-field type control and mean-field games with common noise, with application to production of an exhaustible resource. Applied Mathematics and Optimization, 74(3):459–486, 2016.
J. Huang, X. Li, and J. Yong. Linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control and Related Fields, 5(1):97–139, 2015.
X. Li, J. Sun, and J. Xiong. Linear quadratic optimal control problems for mean-field backward stochastic differential equations. Applied Mathematics & Optimization, to appear, 2017.
H. Pham. Linear quadratic optimal control of conditional mckean-vlasov equation with random coefficients and applications. Probability, Uncertainty and Quantitative Risk, 1:7, 2016.
H. Pham and X. Wei. Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics. SIAM Journal on Control and Optimization, 55(2):1069–1101, 2017.
J. Yong. Linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. SIAM Journal on Control and Optimization, 51(4):2809–2838, 2013.
J. Yong and X.Y. Zhou. Stochastic controls: Hamiltonian Systems and HJB Equations. SMAP. Springer, 1999.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Miller, E., Pham, H. (2019). Linear-Quadratic McKean-Vlasov Stochastic Differential Games. In: Yin, G., Zhang, Q. (eds) Modeling, Stochastic Control, Optimization, and Applications. The IMA Volumes in Mathematics and its Applications, vol 164. Springer, Cham. https://doi.org/10.1007/978-3-030-25498-8_19
Download citation
DOI: https://doi.org/10.1007/978-3-030-25498-8_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-25497-1
Online ISBN: 978-3-030-25498-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)