Stopping Times in the Filtration of a Brownian Motion Stopped at its Last Passage Time

Abstract

We investigate the structural properties of the last passage time ${\sigma }_z^{\lambda }$ at level $z>0$ of a Brownian motion with positive drift $\lambda >0$, denoted $B^{\lambda }=(B_t+\lambda t{)}_{t\ge 0}$, in the filtration generated by the process ${\xi }^{\lambda ,z}=(B_{t\wedge {\sigma }_z^{\lambda }}^{\lambda }{)}_{t\ge 0}$. We compute the compensator of ${\sigma }_z^{\lambda }$ and establish that it is the unique totally inaccessible stopping time in the filtration of ${\xi }^{\lambda ,z}$. Moreover, we provide a canonical decomposition of arbitrary stopping times: for any stopping time $T$, the restriction of $T$ to the set $\{T={\sigma }_z^{\lambda }\}$ is totally inaccessible, while its restriction to $\{T\ne {\sigma }_z^{\lambda }\}$ is predictable. Although the paths of ${\xi }^{\lambda ,z}$ are continuous, the process fails to satisfy the Feller property and is not strong Markov. Nevertheless, we show that its natural filtration is quasi-left-continuous. To overcome these limitations, we consider the extended process ${\zeta }^{\lambda ,z}=({\mathbb{I}}_{\{t<{\sigma }_z^{\lambda }\}},{\xi }_t^{\lambda ,z}{)}_{t\ge 0}$, and prove that it is a Feller process. We compute its infinitesimal generator, which allows us to characterize the associated class of martingales and identify the solutions to certain partial differential equations.