Conditioned Martingales

TL;DR

This paper provides a simple proof that conditioned Brownian motions relate to Bessel processes and extends this understanding to general continuous local martingales, clarifying the role of time in these conditions.

Contribution

It generalizes the known relationship between conditioned Brownian motions and Bessel processes to all continuous local martingales, offering a clearer theoretical framework.

Findings

Upward conditioned Brownian motion is a 3D Bessel process.

Downward conditioned Bessel process is a Brownian motion.

Law of conditioned regular diffusions is characterized.

Abstract

It is well known that upward conditioned Brownian motion is a three-dimensional Bessel process, and that a downward conditioned Bessel process is a Brownian motion. We give a simple proof for this result, which generalizes to any continuous local martingale and clarifies the role of finite versus infinite time in this setting. As a consequence, we can describe the law of regular diffusions that are conditioned upward or downward.