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We analyze both finite and infinite systems of Riccati equations derived from stochastic differential games on infinite networks. We discuss a connection to the Catalan numbers and the convergence of the Catalan functions by Fourier transforms. 

Abstract We analyze the systemic risk for disjoint and overlapping groups of financial institutions by proposing new models with realistic game features.Specifically, we generalize the systemic risk measure proposed in[F. Biagini, J.-P. Fouque, M. Frittelli and T. Meyer-Brandis, On fairness of systemic risk measures, Finance Stoch. 24 (2020), 2, 513–564]by allowing individual banks to choose their preferred groups instead of being assigned to certain groups.We introduce the concept of Nash equilibrium for these new models, and analyze the optimal solution under Gaussian distribution of the risk factor.We also provide an explicit solution for the risk allocation of the individual banks and study the existence and uniqueness of Nash equilibrium both theoretically and numerically.The developed numerical algorithm can simulate scenarios of equilibrium, and we apply it to study the banking structure with real data and show the validity of the proposed model. 

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. 

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.