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 $B^{\lambda} = (B_t + \lambda t)_{t \geq 0}$, in the filtration generated by the process $\xi^{\lambda,z} = (B^{\lambda}_{t \wedge \sigma_z^{\lambda}})_{t \geq 0}$. 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 $\{T \neq \sigma_z^\lambda\}$ 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 $\zeta^{\lambda,z} = (\mathbb{I}_{\{t < \sigma_z^\lambda\}}, \xi_t^{\lambda,z})_{t \geq 0}$, 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.