The last passage time of a drifted Brownian motion is the unique totally inaccessible stopping time in its stopped filtration; the extended process with an indicator of whether time is before the passage is Feller.
A reading guide for last passage times with financial applications in view
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Stopping Times in the Filtration of a Brownian Motion Stopped at its Last Passage Time
The last passage time of a drifted Brownian motion is the unique totally inaccessible stopping time in its stopped filtration; the extended process with an indicator of whether time is before the passage is Feller.