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.
"Conditioned martingales." Electron. Commun. Probab. 17 1 - 12, 2012. https://doi.org/10.1214/ECP.v17-1955