Collision times and exit times from cones: a duality

Abstract

We consider the first collision time for a set of independent one-dimensional zero-drift Wiener processes. For the 3-process problem, the first collision time corresponds to the first exit time of Brownian motion in a cone in R 2, and we can apply the results of Spitzer (1958) and Dante DeBlassie (1987) to obtain its distribution. In the case where the processes have equal infinitesimal variance, a more elementary method yields nice closed-form results for the 3-process problem, and second order approximations for the general n-process problem. This case (for three processes) corresponds to Brownian motion in a cone of angle 1 3 π. The latter approach can in fact be applied to any system of independent (identical) Markov processes, provided the single-barrier hitting time distributions are known for the individual processes and their differences, and provided the processes can't jump over each other.