Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Contagious McKean--Vlasov problems with common noise: from smooth to singular feedback through hitting times

Ben Hambly, Aldaïr Petronilia, Christoph Reisinger, Stefan Rigger, Andreas Søjmark

TL;DR

This work analyzes contagious McKean--Vlasov problems with a common noise source, connecting smoothed, latency-influenced feedback to instantaneous singular interactions. By embedding mollified systems into an $M_1$-topology framework and constructing extended processes, the authors prove weak convergence to relaxed solutions of the singular equation, derive jump-size bounds, and establish conditions for almost-sure convergence to strong, physical solutions in time-homogeneous settings. They also introduce a minimal-solution scheme that yields a $W^0$-measurable limit satisfying a physical jump condition, and provide rates of convergence up to the breakdown of regularity, complemented by numerical experiments that explore convergence orders under various regularity and noise scenarios. The results advance understanding of how latency and contagion in mean-field networks with common shocks transition to instantaneous, jump-driven dynamics and offer practical insights for assessing systemic risk models with abrupt contagion events.

Abstract

We consider a family of McKean--Vlasov equations arising as the large particle limit of a system of interacting particles on the positive half-line with common noise and feedback. Such systems are motivated by structural models for systemic risk with contagion. This contagious interaction is such that when a particle hits zero, the impact is to move all the others toward the origin through a kernel which smooths the impact over time. We study a rescaling of the impact kernel under which it converges to the Dirac delta function so that the interaction happens instantaneously and the limiting singular McKean--Vlasov equation can exhibit jumps. Our approach provides a novel method to construct solutions to such singular problems that allows for more general drift and diffusion coefficients and we establish weak convergence to relaxed solutions in this setting. With more restrictions on the coefficients we can establish an almost sure version showing convergence to strong solutions. Under some regularity conditions on the contagion, we also show a rate of convergence up to the time the regularity of the contagion breaks down. Lastly, we perform some numerical experiments to investigate the sharpness of our bounds for the rate of convergence.

Contagious McKean--Vlasov problems with common noise: from smooth to singular feedback through hitting times

TL;DR

This work analyzes contagious McKean--Vlasov problems with a common noise source, connecting smoothed, latency-influenced feedback to instantaneous singular interactions. By embedding mollified systems into an -topology framework and constructing extended processes, the authors prove weak convergence to relaxed solutions of the singular equation, derive jump-size bounds, and establish conditions for almost-sure convergence to strong, physical solutions in time-homogeneous settings. They also introduce a minimal-solution scheme that yields a -measurable limit satisfying a physical jump condition, and provide rates of convergence up to the breakdown of regularity, complemented by numerical experiments that explore convergence orders under various regularity and noise scenarios. The results advance understanding of how latency and contagion in mean-field networks with common shocks transition to instantaneous, jump-driven dynamics and offer practical insights for assessing systemic risk models with abrupt contagion events.

Abstract

We consider a family of McKean--Vlasov equations arising as the large particle limit of a system of interacting particles on the positive half-line with common noise and feedback. Such systems are motivated by structural models for systemic risk with contagion. This contagious interaction is such that when a particle hits zero, the impact is to move all the others toward the origin through a kernel which smooths the impact over time. We study a rescaling of the impact kernel under which it converges to the Dirac delta function so that the interaction happens instantaneously and the limiting singular McKean--Vlasov equation can exhibit jumps. Our approach provides a novel method to construct solutions to such singular problems that allows for more general drift and diffusion coefficients and we establish weak convergence to relaxed solutions in this setting. With more restrictions on the coefficients we can establish an almost sure version showing convergence to strong solutions. Under some regularity conditions on the contagion, we also show a rate of convergence up to the time the regularity of the contagion breaks down. Lastly, we perform some numerical experiments to investigate the sharpness of our bounds for the rate of convergence.
Paper Structure (17 sections, 35 theorems, 254 equations, 5 figures, 5 tables, 1 algorithm)

This paper contains 17 sections, 35 theorems, 254 equations, 5 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Weak convergence of smoothed feedback systems
  3. Limit points of the smoothed system
  4. Continuity of hitting times
  5. Martingale arguments and convergence
  6. Stronger mode of convergence
  7. Properties of $\Gamma$ and existence of strong solutions
  8. Convergence of minimal solutions
  9. Rates of convergence
  10. Theoretical estimates on rates of convergence
  11. Numerical simulations
  12. Initial density vanishing at zero and no discontinuity or common noise
  13. Setting with discontinuity and without common noise
  14. Simulations with common noise
  15. Technical lemmas
  16. ...and 2 more sections

Key Result

Theorem 2.3

There is a unique strong solution to the SPDE where the coefficients $b,\,\sigma$, $\rho,\,\kappa^\varepsilon$, and $\alpha$ satisfy Assumption ass: MODIFIED AND SIMPLIED FROM SOJMARK SPDE PAPER ASSUMPTIONS II, and $\phi \in \mathscr{C}_0$, the set of Schwartz functions that are zero at $0$.

Figures (5)

  • Figure 1: Initial density vanishing at zero with no discontinuity or common noise
  • Figure 2: Initial density vanishing at zero with discontinuity and no common noise
  • Figure 3: Initial density vanishing at zero with common noise
  • Figure 4: Sensitivity of rate of convergence with respect to changes in $\Delta_t$
  • Figure 5: Sensitivity of the rate of convergence with respect to changes in $\Delta_t$

Theorems & Definitions (70)

  • Definition 2.1: Relaxed solutions
  • Theorem 2.3: hambly2019spde
  • Theorem 2.4: hambly2019spde
  • Theorem 2.5: Existence and convergence generalised
  • Corollary 2.6: Existence of physical solutions
  • Remark 2.7
  • Proposition 2.8: Grönwall upper bound
  • proof
  • Proposition 2.9: Tightness of smoothed random measures
  • proof
  • ...and 60 more