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 $σ_z^λ$ at level $z > 0$ of a Brownian motion with positive drift $λ> 0$, denoted $B^λ = (B_t + λt)_{t \geq 0}$, in the filtration generated by the process $ξ^{λ,z} = (B^λ_{t \wedge σ_z^λ})_{t \geq 0}$. We compute the compensator of $σ_z^λ$ and establish that it is the unique totally inaccessible stopping time in the filtration of $ξ^{λ,z}$. Moreover, we provide a canonical decomposition of arbitrary stopping times: for any stopping time $T$, the restriction of $T$ to the set $\{T = σ_z^λ\}$ is totally inaccessible, while its restriction to $\{T \neq σ_z^λ\}$ is predictable. Although the paths of $ξ^{λ,z}$ 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 $ζ^{λ,z} = (\mathbb{I}_{\{t < σ_z^λ\}}, ξ_t^{λ,z})_{t \geq 0}$, 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.