Conditioned Martingales
arXiv:1203.2587 · doi:10.1214/ECP.v17-1955
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.
Corrected several typos, improved formulations. Accepted by Electronic Communications in Probability; Electronic Communications in Probability, 2012, Volume 17, Issue 48