Zhongyuan Cao

We study continuous stochastic games with heterogeneous mean field interactions and jumps on large networks and explore their limit counterparts. We introduce the graphon game model based on a controlled graphon mean field stochastic differential equation system with jumps, which can be regarded as the limiting case of a finite game dynamic system as the number of players tends to infinity. We examine the case of controlled dynamics, with control terms present in the drift, diffusion, and jump components. We focus on the study of the limit theory and provide convergence results on the state trajectories and their laws, transitioning from finite game systems to graphon systems. We also study approximate equilibria for finite games on large networks, using the graphon equilibrium as a benchmark. The rates of convergence are analyzed under various underlying graphon models and regularity assumptions.

In this paper, we provide central limit theorems to analyze the combined effects of fire sales and default cascades on systemic risk within stochastic financial networks. We introduce a series of central limit theorems that explore the dynamics of total shares sold and the equilibrium pricing of illiquid assets within a stylized fire sales framework. We show that the equilibrium prices of these assets have asymptotically Gaussian fluctuations. In our numerical experiments, we demonstrate how our central limit theorems can be applied to construct confidence intervals for the magnitude of contagion and the extent of losses due to fire sales.

We consider a general tractable model for default contagion and systemic risk in a heterogeneous financial network subjected to an exogenous macroeconomic shock. We show that under certain regularity assumptions, the default cascade model can be transformed into a death process problem represented by a balls-and-bins model. We state various limit theorems regarding the final size of default cascades. Under appropriate assumptions on the degree and threshold distributions, we prove that the final sizes of default cascades have asymptotically Gaussian fluctuations. We next state limit theorems for different system-wide wealth aggregation functions, which enable us to provide systemic risk measures in relation to the structure and heterogeneity of the financial network. Lastly, we demonstrate how these results can be utilized by a social planner to optimally target interventions during a financial crisis given a budget constraint and under partial information of the financial network.

We introduce and examine a default cascade process within stochastic financial networks. This process involves a finite set of agents who hold claims against each other. These agents interact pairwise with their counterparties at random times, which are determined by a Poisson process, and during these interactions, they update their states. If the debtor agent is solvent at the time of the meeting, the pair continue to interact in future encounters. However, if the debtor has defaulted, the creditor incurs a random loss, the distribution of which depends on the creditor’s characteristics. Following such an event, the two agents cease future interactions. First, we simplify the problem by classifying banks into different types. Machine learning techniques can be utilized to calibrate these types to real-world data. Nodes can be assigned attributes based on a variety of financial factors or characteristics. Our principal result is a precise asymptotic expression for the proportion of defaulted agents at any given time. This is particularly relevant in the case of heterogeneous random financial networks, where agents interact with counterparties within a sparse directed random graph and meeting times are i.i.d. exponential random variables.