A note about conditional Ornstein-Uhlenbeck processes

TL;DR

This paper explores the relationship between a killed Ornstein-Uhlenbeck process and a conditioned radial Ornstein-Uhlenbeck process, deriving an identity in law through a straightforward absolute continuity approach.

Contribution

It provides a new derivation of an existing law identity between two Ornstein-Uhlenbeck processes using simple absolute continuity methods.

Findings

Established the law identity between the two processes.

Demonstrated the use of absolute continuity in process relationships.

Simplified the derivation of known stochastic process identities.

Abstract

In this short note, the identity in law, which was obtained by P. Salminen, between on one hand, the Ornstein-Uhlenbeck process with parameter gamma, killed when it reaches 0, and on the other hand, the 3-dimensional radial Ornstein-Uhlenbeck process killed exponentially at rate gamma and conditioned to hit 0, is derived from a simple absolute continuity relationship.