On collisions of Brownian particles

TL;DR

This paper investigates the conditions under which multiple Brownian particles with variable drift and diffusion collide or avoid collisions, applying findings to a financial market model with reflecting boundaries.

Contribution

It provides new sufficient conditions for the occurrence or avoidance of triple collisions in systems of Brownian particles with configuration-dependent coefficients, including applications to market models.

Findings

Established criteria for triple collision occurrence or avoidance.

Applied collision analysis to a reflected Brownian motion model in finance.

Provided insights into particle interactions with configuration-dependent dynamics.

Abstract

We examine the behavior of $n$ Brownian particles diffusing on the real line with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the current configuration of particles. Sufficient conditions are established for the absence and for the presence of triple collisions among the particles. As an application to the Atlas model for equity markets, we study a special construction of such systems of diffusing particles using Brownian motions with reflection on polyhedral domains.