A Moving Boundary Problem for Brownian Particles with Singular Forward-Backward Interactions

TL;DR

This work analyzes a system of $N$ Brownian particles with associated moving boundaries defined by conditional absorption probabilities, creating a singular forward-backward SDE coupled through hitting times. The authors develop a cascade of moving-boundary PDEs whose decoupling fields provide a global solution strategy, prove classical well-posedness and continuity of the PDE system, and then use the verification theorem to construct a corresponding FBSDE solution and establish its uniqueness. A threefold singularity arises from hitting-time coupling, discontinuous terminal data, and degenerate diffusion upon absorption, which the authors address via the cascade PDE framework and fixed-point arguments. The paper also outlines a mean-field limit, showing potential nonuniqueness in the limit and highlighting connections to contagion dynamics in financial networks. Overall, the approach advances the theory of singular FBSDEs by embedding absorption-driven moving-boundary PDEs as a robust decoupling mechanism with rigorous existence, uniqueness, and connections to mean-field behavior.

Abstract

We introduce a system of Brownian particles, each absorbed upon hitting an associated moving boundary. The boundaries are determined by the conditional probabilities of the particles being absorbed before some final time horizon, given the current knowledge of the system. While the particles evolve forward in time, the conditional probabilities are computed backwards in time, leading to a specification of the particle system as a system of singular forward-backward SDEs coupled through hitting times. Its analysis leads to a novel type of tiered moving boundary problem. Each level of this PDE corresponds to a different configuration of unabsorbed particles, with the boundary and the boundary condition of a given level being determined by the solution of the preceding one. We establish classical well-posedness for this moving boundary problem and use its solution to solve the original forward-backward system and prove its uniqueness.

Figures (2)

• Figure 1: An idealised picture of two particles (dotted lines) and their moving boundaries (full lines) with $D_{12}=D_{21}=1$ and $D_{11}=D_{22}=0$. At $\tau_2$, the $2$nd particle (orange, circle markers) crosses the boundary and is absorbed, so the boundary of the $1$st particle (purple, star markers) settles at $1$ from then on. The $1$st particle is not absorbed, so the boundary of the $2$nd particle instead ends up at $0$ (as opposed to $1$).
• Figure 2: The plots show the conditional killing probability for a system with two particles. The left-hand side shows the probability if one particle was already removed, while on the right-hand side both particles are initially alive.

Theorems & Definitions (39)

• Remark 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Remark 2.4
• Theorem 2.5
• proof
• Corollary 2.6
• proof
• Proposition 2.7