系统性风险与异质性平均场类型银行网络
Systemic Risk and Heterogeneous Mean Field Type Interbank Network
AI总结：
本文研究了基于相对对数资本化的异质性银行借贷系统，通过耦合扩散方程描述对数资本化的演化，并探讨了在两个银行组中存在充分大数量银行时，耦合里卡蒂方程的可解性保证了均衡的存在，同时分析了异质性平均场博弈在任意数量组中的存在性以及参数对流动性率的影响。
摘要：
我们研究了基于相对对数资本化的异质性银行借贷和借贷系统，该系统由组内平均值和总体平均值的线性组合给出的相对平均值构成，并通过耦合扩散方程描述对数资本化的演化。该模型包含一个具有组内同质性和组间异质性的博弈特征，其中银行通过最小化异质性线性二次成本来寻找最优借贷或借贷策略，以避免接近违约障碍。由于借贷和借贷系统的复杂性，封闭环纳什均衡和开放环纳什均衡均由耦合里卡蒂方程驱动。在两个银行组的情况下，当银行数量足够大时，均衡的存在性由耦合里卡蒂方程在每组中随银行数量趋于无穷时的可解性保证。均衡由与单组博弈相同的均值回复项和由于异质性导致的组平均组成。此外，还讨论了任意数量组的异质性平均场博弈。在一般情况下，对于任意数量的异质组，也验证了ε-纳什均衡的存在性。最后，在金融含义方面，我们观察到由均值回复项和个体组总体平均值的线性组合所支配的纳什均衡，并通过数值分析研究了相对参数对流动性率的影响。
英文摘要：
We study the system of heterogeneous interbank lending and borrowing based on the relative average of log-capitalization given by the linear combination of the average within groups and the ensemble average and describe the evolution of log-capitalization by a system of coupled diffusions. The model incorporates a game feature with homogeneity within groups and heterogeneity between groups where banks search for the optimal lending or borrowing strategies through minimizing the heterogeneous linear quadratic costs in order to avoid to approach the default barrier. Due to the complicity of the lending and borrowing system, the closed-loop Nash equilibria and the open-loop Nash equilibria are both driven by the coupled Riccati equations. The existence of the equilibria in the two-group case where the number of banks are sufficiently large is guaranteed by the solvability for the coupled Riccati equations as the number of banks goes to infinity in each group. The equilibria are consisted of the mean-reverting term identical to the one group game and the group average owing to heterogeneity. In addition, the corresponding heterogeneous mean filed game with the arbitrary number of groups is also discussed. The existence of the $ε$-Nash equilibrium in the general $d$ heterogeneous groups is also verified. Finally, in the financial implication, we observe the Nash equilibria governed by the mean-reverting term and the linear combination of the ensemble averages of individual groups and study the influence of the relative parameters on the liquidity rate through the numerical analysis.
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