We consider vector fixed point (FP) equations in large dimensional spaces involving random variables, and study their realization-wise solutions. We have an underlying directed random graph, that defines the connections between various components of the FP equations. Existence of an edge between nodes i, j implies the i th FP equation depends on the j th component. We consider a special case where any component of the FP equation depends upon an appropriate aggregate of that of the random neighbor components. We obtain finite dimensional limit FP equations (in a much smaller dimensional space), whose solutions approximate the solution of the random FP equations for almost all realizations, in the asymptotic limit (number of components increase). Our techniques are different from the traditional mean-field methods, which deal with stochastic FP equations in the space of distributions to describe the stationary distributions of the systems. In contrast our focus is on realization-wise FP solutions. We apply the results to study systemic risk in a large financial heterogeneous network with many small institutions and one big institution, and demonstrate some interesting phenomenon.
Asymptotics of Systemic Risk in a Renewal Model with Multiple Business Lines and Heterogeneous Claims
School of Big Data and Statistics, Anhui University（安徽大学大数据与统计学院）；School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理工学院）
Linear-quadratic optimal control for non-exchangeable mean-field SDEs and applications to systemic risk
Université Paris Cité（巴黎西岱大学）；Sorbonne University（索邦大学）；Institut für Mathematik, Technische Universität Berlin（柏林工业大学数学研究所）；Luiss Guido Carli University（鲁伊西斯·圭多·卡利大学）；Ecole Polytechnique（巴黎综合理工学院）
