Paper: arXiv 2212.08518

Authors: Erhan Bayraktar, Gaoyue Guo, Wenpin Tang, Yuming Paul Zhang

Abstract

This paper is concerned with the problem of budget control in a large particle system modeled by stochastic differential equations involving hitting times, which arises from considerations of systemic risk in a regional financial network. Motivated by Tang and Tsai (Ann. Probab., 46(2018), pp. 1597{1650), we focus on the number or proportion of surviving entities that never default to measure the systemic robustness. First we show that both the mean-field particle system and its limiting McKean-Vlasov equation are well-posed by virtue of the notion of minimal solutions. We then establish a connection between the proportion of surviving entities in the large particle system and the probability of default in the limiting McKean-Vlasov equation as the size of the interacting particle system N tends to infinity. Finally, we study the asymptotic efficiency of budget control in different economy regimes: the expected number of surviving entities is of constant order in a negative economy; it is of order of the square root of N in a neutral economy; and it is of order N in a positive economy where the budget’s effect is negligible.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using mean-field particle systems, McKean-Vlasov SDEs, and hitting times to model systemic risk and budget control. However, it is entirely theoretical with no empirical backtesting, market data validation, or provided code.

Research Flowchart

  flowchart TD
    A[Research Goal: Evaluate systemic robustness & budget control efficiency in financial networks] --> B[Model: SDE Interacting Particle System with Hitting Times]
    B --> C[Establish Minimal Solutions for Well-Posedness of Particle System & McKean-Vlasov Limit]
    C --> D[Prove Propagation of Chaos: Connect N-particle survival proportion to limit default probability as N → ∞]
    D --> E[Analyze Budget Control Efficiency Across Economy Regimes]
    E --> F[Findings: Expected Surviving Entities<br>- Negative Economy: O1<br>- Neutral Economy: O√N<br>- Positive Economy: ON]