Approximating diffusion reflections at elastic boundaries

Dirk Becherer,Todor Bilarev,Peter Frentrup

doi:10.1214/18-ecp141

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.

Highlights

The classical Skorokhod problem is that of reflecting a path at a boundary
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

Summary

Introduction

The classical Skorokhod problem is that of reflecting a path at a boundary. It is a standard tool to construct solutions to SDEs with reflecting boundary conditions. Such elastic boundaries appear typically in solutions to singular control problems of finite fuel type, where the optimal control is the reflection local time that keeps a diffusion process within a no-action region, cf Karatzas and Shreve [5]. A natural idea for approximation is to proxy ’infinitesimal’ reflections by small ε-jumps ∆Lε, thereby inducing jumps of the elastic reflection boundary, see Figure 2 This allows to express excursion lengths of the approximating diffusion Xε in terms of independent hitting times for continuous diffusions, what naturally leads to an explicit expression (3.9) for the Laplace transform of the inverse local time of X. We prove ucp-convergence of (Xε, Lε) to (X, L) by showing in Section 4 tightness of the approximation sequence (Xε, Lε)ε and using Kurtz–Protter’s notion of uniformly controlled variations (UCV), introduced in [8]

Elastic reflection: model and notation

Approximation by small ε-reflections

Tightness and convergence