Chris Mount: Exact Bridges for Reflected Sticky Brownian Motion

Reflected sticky Brownian motion is the canonical example of a stochastic process that spends positive time at the boundary without remaining there throughout any time interval. It arises from one of the boundary conditions in Feller's classification of one-dimensional diffusions, and it is the simplest model of a particle that displays nontrivial boundary behavior. A classical result makes it easy to sample the process exactly at increasing times. We seek to answer a simple question: given the process at two times, how do we sample it in between? For Brownian motion and reflected Brownian motion, the answer is the Brownian bridge. We will show that for reflected sticky Brownian motion, the right move is to track the local time of the process. This yields exact samples of the process at any finite set of times, queried in any order.