Pinned diffusions and Markov bridges

TL;DR

This paper investigates conditions under which certain diffusion processes are 'pinned' at a point and explores whether these can be represented as bridges of Gaussian Markov processes or Itô diffusions, providing concrete examples.

Contribution

It provides a simple criterion for pinning diffusions and characterizes when they can be viewed as bridges of Gaussian Markov processes or Itô diffusions.

Findings

Established a checkable condition for diffusion pinning.

Characterized when pinned diffusions are Gaussian or Itô bridges.

Provided illustrative examples of the theoretical results.

Abstract

In this article we consider a family of real-valued diffusion processes on the time interval $[0,1]$ indexed by their prescribed initial value $x \in \mathbb{R}$ and another point in space, $y \in \mathbb{R}$. We first present an easy-to-check condition on their drift and diffusion coefficients ensuring that the diffusion is pinned in $y$ at time $t=1$. Our main result then concerns the following question: can this family of pinned diffusions be obtained as the bridges either of a Gaussian Markov process or of an It\^o diffusion? We eventually illustrate our precise answer with several examples.