Approximating diffusion reflections at elastic boundaries

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 $\epsilon$-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.