Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2605.14254 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917494082502656 |
|---|---|
| author | Louriki, Mohammed |
| author_facet | Louriki, Mohammed |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14254 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stopping Times in the Filtration of a Brownian Motion Stopped at its Last Passage Time Louriki, Mohammed Probability 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. |
| title | Stopping Times in the Filtration of a Brownian Motion Stopped at its Last Passage Time |
| topic | Probability |
| url | https://arxiv.org/abs/2605.14254 |