arxiv: 2605.14254 · v1 · submitted 2026-05-14 · 🧮 math.PR

Recognition: 2 theorem links

· Lean Theorem

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

Mohammed Louriki

Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:39 UTC · model grok-4.3

classification 🧮 math.PR

keywords Brownian motion with driftlast passage timetotally inaccessible stopping timecompensatorstopped processquasi-left-continuous filtrationFeller process

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The pith

The last passage time of Brownian motion with positive drift is the unique totally inaccessible stopping time in the filtration of the stopped process.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the last passage time at a fixed level for a Brownian motion with positive drift, considered inside the filtration generated by the same motion after it has been stopped at that time. It derives the compensator of this last passage time and shows that the time is the only totally inaccessible stopping time present in the filtration. The work further decomposes every stopping time according to whether or not it coincides with the last passage time, assigning the inaccessible part only to the coincidence set. Although the stopped process has continuous paths, it lacks the strong Markov and Feller properties, yet its filtration remains quasi-left-continuous. An auxiliary two-component process that records whether the last passage time has occurred restores the Feller property and yields an explicit generator for martingale and PDE analysis.

Core claim

In the filtration generated by the process ξ^λ,z obtained by stopping a Brownian motion with positive drift λ at its last passage time σ_z^λ above level z, the random time σ_z^λ admits an explicit compensator and is the unique totally inaccessible stopping time. For an arbitrary stopping time T, the restriction of T to the set {T = σ_z^λ} is totally inaccessible while the restriction to the complementary set is predictable. The natural filtration of ξ^λ,z is quasi-left-continuous even though the process itself fails to be strong Markov or Feller; the auxiliary process ζ^λ,z that augments ξ^λ,z with the indicator of whether the last passage time has occurred is Feller, and its generator fully

What carries the argument

The last passage time σ_z^λ together with its compensator inside the filtration of the stopped drifted Brownian motion ξ^λ,z; the auxiliary Feller process ζ^λ,z records the occurrence of σ_z^λ and supplies the generator.

If this is right

Every stopping time admits a canonical splitting into a totally inaccessible part supported only on {T = σ_z^λ} and a predictable part supported on the complement.
Martingales in the filtration of ξ^λ,z can be constructed from the compensator of σ_z^λ.
The quasi-left-continuity of the filtration guarantees that every martingale has left limits that coincide with predictable projections.
Solutions to certain parabolic PDEs are characterized via the generator of the extended Feller process ζ^λ,z.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same compensator calculation may apply to last passage times of Lévy processes with positive drift.
The decomposition supplies a concrete example of a filtration that is quasi-left-continuous without the underlying process being strong Markov.
The generator of ζ^λ,z offers a route to explicit solutions of optimal-stopping problems that involve the last passage time.

Load-bearing premise

The analysis uses only the standard construction of Brownian motion with positive drift and the usual right-continuous augmentation of the filtration generated by the stopped continuous-path process.

What would settle it

An explicit construction of a stopping time T distinct from σ_z^λ whose compensator vanishes on a set of positive probability would contradict the claimed uniqueness of the totally inaccessible time.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper computes the compensator of the last passage time in this stopped filtration, proves it is the unique totally inaccessible stopping time there, and gives a decomposition rule for arbitrary stopping times plus a Feller extension of the process.

read the letter

The main thing to know is that the paper works out the compensator of σ_z^λ inside the filtration generated by the stopped drifted Brownian motion ξ^{λ,z} and shows this stopping time is the only totally inaccessible one in that filtration. It also gives a clean decomposition: any stopping time T splits into a totally inaccessible piece on the set where it equals σ and a predictable piece elsewhere. To fix the fact that ξ is continuous but neither Feller nor strong Markov, they add the indicator process to create ζ and prove that ζ is Feller, then compute its generator explicitly. These statements are new relative to the usual literature on Brownian filtrations and last passage times, and they rest on standard properties of Brownian motion with positive drift without circular definitions or fitted parameters. The generator calculation is concrete enough to characterize the associated martingales and link to some PDEs. The stress-test note found no internal inconsistencies in moving from the stopped continuous process to the jump-augmented Feller process, which matches what the abstract lays out. The soft spots are limited. The results stay inside a narrow corner of stochastic processes, so the payoff is mainly for people already working on stopped filtrations rather than a broader shift in the field. Without the full derivations in front of me I cannot check every step of the quasi-left-continuity argument, but the classical toolkit used here makes large gaps unlikely. The assumptions are the usual ones for drifted Brownian motion and its natural filtration. This is for specialists who care about stopping times and filtrations generated by paths stopped at last passage times. A reader already thinking about decompositions or Feller properties in similar settings would get direct value from the explicit formulas. I would send it to peer review. The claims are precise, the setup is standard, and the new pieces are stated clearly enough to benefit from referee input.

Referee Report

0 major / 3 minor

Summary. The paper investigates structural properties of the last passage time σ_z^λ of a Brownian motion with positive drift λ > 0 in the filtration generated by the stopped process ξ^{λ,z}. It computes the compensator of σ_z^λ and proves it is the unique totally inaccessible stopping time therein; for arbitrary stopping times T it decomposes the restriction to {T = σ_z^λ} as totally inaccessible and to {T ≠ σ_z^λ} as predictable. Although ξ^{λ,z} has continuous paths but is neither Feller nor strong Markov, the natural filtration is shown to be quasi-left-continuous. The authors introduce the augmented process ζ^{λ,z} = (1_{t < σ_z^λ}, ξ_t^{λ,z}), prove it is Feller, compute its infinitesimal generator, and use the generator to characterize associated martingales and solve certain PDEs.

Significance. If the derivations hold, the work supplies explicit compensator formulas and a canonical decomposition of stopping times in a filtration that is quasi-left-continuous yet generated by a non-Markov process. The construction of the Feller process ζ^{λ,z} together with its generator computation furnishes a concrete analytic tool for martingale problems and PDEs that would otherwise be inaccessible; this extends classical last-passage-time results for drifted Brownian motion and may be useful in optimal stopping or stochastic control settings.

minor comments (3)

[§2] §2: the definition of the augmented filtration for ξ^{λ,z} should explicitly state whether right-continuity is imposed before or after the compensator calculation, to avoid ambiguity in the uniqueness argument.
[§4] §4, generator computation: the domain of the generator on ζ is described via test functions vanishing at the boundary; a brief verification that these functions are dense in the appropriate space would strengthen the Feller property claim.
[Notation] Notation: the indicator process in ζ^{λ,z} is written both as I and as 1_{t<σ}; adopt a single symbol throughout.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and accurate summary of our manuscript, as well as for the positive assessment of its significance and potential applications. We are pleased that the explicit compensator formulas, the canonical decomposition of stopping times, the quasi-left-continuity of the filtration, and the construction of the Feller process ζ^{λ,z} together with its generator are recognized as useful contributions.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivations rest entirely on classical properties of Brownian motion with positive drift, the standard augmentation of its natural filtration, and the explicit construction of the auxiliary Feller process ζ^{λ,z} together with its generator. The compensator computation for σ_z^λ, its uniqueness as the sole totally inaccessible stopping time, and the predictable/inaccessible decomposition of arbitrary stopping times T are obtained directly from these constructions and the quasi-left-continuity argument; none of the central claims reduce by definition or by self-citation to the target results themselves. No fitted parameters, ansatzes smuggled via prior work, or renaming of known empirical patterns appear.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies exclusively on the standard axioms of Brownian motion with drift and the usual definitions of stopping times and filtrations; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)

standard math Brownian motion with positive drift has continuous paths, independent increments, and the strong Markov property in its natural filtration.
Invoked implicitly throughout the construction of ξ^λ,z and the analysis of its stopping times.

pith-pipeline@v0.9.0 · 5577 in / 1355 out tokens · 33356 ms · 2026-05-15T02:39:21.131593+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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}.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat ≃ Nat unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the extended process ζ^{λ,z} = (I_{t<σ_z^λ}, ξ_t^{λ,z}) is a Feller process. We compute its infinitesimal generator

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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