{"paper":{"title":"Stopping Times in the Filtration of a Brownian Motion Stopped at its Last Passage Time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The last passage time of Brownian motion with positive drift is the unique totally inaccessible stopping time in the filtration of the stopped process.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Mohammed Louriki","submitted_at":"2026-05-14T01:47:48Z","abstract_excerpt":"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 $T$, th"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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, for any stopping time T, the restriction of T to {T = σ_z^λ} is totally inaccessible, while its restriction to {T ≠ σ_z^λ} is predictable.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis assumes the standard construction of Brownian motion with positive drift λ > 0 and the usual augmentation of the natural filtration generated by the stopped process ξ^λ,z; no additional regularity beyond continuity of paths is postulated.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The last passage time of Brownian motion with positive drift is the unique totally inaccessible stopping time in the filtration of the stopped process.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b0d296ca7f80607a05245ad5bd95dddaa8d5bd3cf75b373e76eecbfb11055817"},"source":{"id":"2605.14254","kind":"arxiv","version":1},"verdict":{"id":"3eb412f7-1c39-45e7-8216-74991bb4473a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:39:21.131593Z","strongest_claim":"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, for any stopping time T, the restriction of T to {T = σ_z^λ} is totally inaccessible, while its restriction to {T ≠ σ_z^λ} is predictable.","one_line_summary":"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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis assumes the standard construction of Brownian motion with positive drift λ > 0 and the usual augmentation of the natural filtration generated by the stopped process ξ^λ,z; no additional regularity beyond continuity of paths is postulated.","pith_extraction_headline":"The last passage time of Brownian motion with positive drift is the unique totally inaccessible stopping time in the filtration of the stopped process."},"references":{"count":52,"sample":[{"doi":"","year":1992,"title":"Abramowitz, M; Stegun I.A.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1992","work_id":"df570099-9f36-426d-b037-5378e4a99dea","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"SpringerBriefs in Quantitative Finance","work_id":"00f6a411-f7e4-479d-b1b9-bc8863152cc4","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"Integral representations of martingales for progres- sive enlargements of filtrations","work_id":"39259871-e8ab-49f6-83ab-f53b4f77bbae","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"On the distribution of the maximum of a Gaussian field withd parameters","work_id":"5cd264eb-256f-43a0-be5a-5ccdbdef9c13","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"L.; Buckdahn, R.; Engelbert, H","work_id":"58c8e71c-8118-4636-8973-7806ebf639f3","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":52,"snapshot_sha256":"136eec2de04872f6ea58859828bc0407d16679449f0451dd5bca5353786ff538","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d0a838084676955933440e830c04a8ca01cb1153f6a41f4395546c551feda3eb"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}