Approximating diffusion reflections at elastic boundaries

TL;DR

This paper establishes a probabilistic limit for one-dimensional diffusions reflected at elastic boundaries, providing a way to approximate local times and their inverse Laplace transforms, relevant for stochastic control problems.

Contribution

It introduces a novel limit result for diffusions reflected at elastic boundaries and offers a new approximation method for reflection local times.

Findings

Derived the Laplace transform of the inverse local time.

Established a limit process for discretely reflected diffusions.

Applicable to finite fuel stochastic control problems.

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 $\varepsilon$-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.