Approximating diffusion reflections at elastic boundaries

Abstract

We show a probabilistic functional limit result for one-dimensional diffusion processes that are reflected at an elastic boundary which is a function of the reflection local time. Such processes are constructed as limits of a sequence of diffusions which are discretely reflected by small jumps at an elastic boundary, with reflection local times being approximated by ε-step processes. The construction yields the Laplace transform of the inverse local time for reflection. Processes and approximations of this type play a role in finite fuel problems of singular stochastic control.

An interacting particle system for the front of an epidemic advancing through a susceptible population

“…In the existing literature, the interacting particle system that is closest to the one we study here is the following system considered in [Bar20]: it consists of n independent Brownian motions (in one dimesion) reflected off a moving boundary, which moves at a speed proportional to the sum of the local times of each Brownian motion along this boundary. Another interesting work in this direction is [BBF18a], which studies a (one-dimensional) SDE reflected off a moving boundary which is a function of the local time of the SDE along the boundary itself (see also [BBF18b] for a financial application of this). These works, however, consider particles that are globally reflected, thus making the analysis quite different from what is needed to handle the notion of infection in our system, which is what drives the moving boundary and determines the effective rate of infection.…”