UC Santa Barbara

In the first part of this thesis, we study linear-quadratic stochastic differential games on directed chains inspired by the directed chain stochastic differential equations introduced by Detering, Fouque and Ichiba [1]. 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. We then continue the analysis through 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. The discussion about stochastic differential games is extended to a random two-sided directed chain and a random directed tree structure.The second part is about systemic risk measures introduced by Biagini, Fouque, Frittelli, and Meyer-Brandis [2]. We first analyze the systemic risk measures for disjoint and overlapping groups (e.g., central clearing counterparties (CCP)) by proposing new models with realistic game features. Specifically, we generalize the systemic risk measure proposed in [2] 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 bank-CCP structure with real data and show the validity of the proposed model.

Under the framework in Biagini, Fouque, Frittelli, and Meyer-Brandis [2], systemic risk measures can be interpreted as the minimal amount of cash that secures the aggregated system by allocating capital to the single institutions before aggregating the individual risks. This problem has no explicit solution except in very limited situations. We then apply a deep learning method as a tool to compute the optimal strategy of capital allocations for the risk measures. Deep learning is increasingly receiving attention in financial modelings and risk management and we propose our deep learning based algorithms to solve both the primal and dual problems of the risk measures, and thus to learn the fair risk allocations. In particular, our method for the dual problem involves the training philosophy inspired by the well-known Generative Adversarial Networks (GAN) approach and a newly designed direct estimation of Radon-Nikodym derivative. In the end, we show substantial numerical studies of the subject and provide interpretations of the risk allocations associated with the systemic risk measures. In the particular case of exponential preferences, numerical experiments demonstrate excellent performance of the proposed algorithm, when compared with the optimal explicit solution as a benchmark.