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1. Linear-Quadratic Stochastic Differential Games on Random Directed Networks

   Feng, Yichen; Fouque, Jean-Pierre; Ichiba, Tomoyuki (January 2021, Journal of mathematics and statistical science)

   null (Ed.)

   The study of linear-quadratic stochastic differential games on directed networks was initiated in Feng, Fouque & Ichiba [7]. In that work, the game on a directed chain with finite or infinite players was defined as well as the game on a deterministic directed tree, and their Nash equilibria were computed. The current work continues the analysis by first developing a random directed chain structure by assuming the interaction between every two neighbors is random. We solve explicitly for an open-loop Nash equilibrium for the system and we find that the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain introduced in [7]. The discussion about stochastic differential games is extended to a random two-sided directed chain and a random directed tree structure. 

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2. Linear-Quadratic Stochastic Differential Games on Directed Chain Networks

   Yichen Feng, Jean-Pierre Fouque (January 2020, Journal of mathematics and statistical science)

   We study linear-quadratic stochastic differential games on directed chains inspired by the directed chain stochastic differential equations introduced by Detering, Fouque and Ichiba. We solve explicitly for Nash equilibria with a finite number of players and we study more general finite-player games with a mixture of both directed chain in- teraction and mean field interaction. We investigate and compare the corresponding games in the limit when the number of players tends to infinity. The limit is characterized by Catalan functions and the dy- namics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain, with or without the presence of mean field interaction. 

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3. Linear-Quadratic Stochastic Differential Games on Directed Chain Networks

   Feng, Yichen; Fouque, Jean-Pierre; Ichiba, Tomoyuki (January 2021, Journal of mathematics and statistical science)

   null (Ed.)

   We study linear-quadratic stochastic differential games on directed chains inspired by the directed chain stochastic differential equations introduced by Detering, Fouque and Ichiba. We solve explicitly for Nash equilibria with a finite number of players and we study more general finite-player games with a mixture of both directed chain interaction and mean field interaction. We investigate and compare the corresponding games in the limit when the number of players tends to infinity. The limit is characterized by Catalan functions and the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain, with or without the presence of mean field interaction. 

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4. Mean-field linear-quadratic stochastic differential games in an infinite horizon

   https://doi.org/10.1051/cocv/2021078  

   Li, Xun; Shi, Jingtao; Yong, Jiongmin (January 2021, ESAIM: Control, Optimisation and Calculus of Variations)

   This paper is concerned with two-person mean-field linear-quadratic non-zero sum stochastic differential games in an infinite horizon. Both open-loop and closed-loop Nash equilibria are introduced. The existence of an open-loop Nash equilibrium is characterized by the solvability of a system of mean-field forward-backward stochastic differential equations in an infinite horizon and the convexity of the cost functionals, and the closed-loop representation of an open-loop Nash equilibrium is given through the solution to a system of two coupled non-symmetric algebraic Riccati equations. The existence of a closed-loop Nash equilibrium is characterized by the solvability of a system of two coupled symmetric algebraic Riccati equations. Two-person mean-field linear-quadratic zero-sum stochastic differential games in an infinite horizon are also considered. Both the existence of open-loop and closed-loop saddle points are characterized by the solvability of a system of two coupled generalized algebraic Riccati equations with static stabilizing solutions. Mean-field linear-quadratic stochastic optimal control problems in an infinite horizon are discussed as well, for which it is proved that the open-loop solvability and closed-loop solvability are equivalent. 

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5. A Probabilistic Approach to Extended Finite State Mean Field Games

   Carmona, Rene; Wang, Peiqi (August 2018, ArXivorg)

   We develop a probabilistic approach to continuous-time finite state mean field games. Based on an alternative description of continuous-time Markov chain by means of semimartingale and the weak formulation of stochastic optimal control, our approach not only allows us to tackle the mean field of states and the mean field of control in the same time, but also extend the strategy set of players from Markov strategies to closed-loop strategies. We show the existence and uniqueness of Nash equilibrium for the mean field game, as well as how the equilibrium of mean field game consists of an approximative Nash equilibrium for the game with finite number of players under different assumptions of structure and regularity on the cost functions and transition rate between states. 

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