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arxiv: 2604.03125 · v1 · submitted 2026-04-03 · 🧮 math.PR

First Passage through a Continuous Barrier: Pathwise Decomposition, Random-Time Structure, and Compensators

Tristan Guillaume (CYU) This is my paper

Pith reviewed 2026-05-13 18:41 UTC · model grok-4.3

classification 🧮 math.PR MSC 60G40
keywords first passage timecontinuous barrierpathwise decompositionpredictabilitycompensatorscàdlàg processesjump diffusionsstopping times
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The pith

The first passage time through a continuous barrier decomposes pathwise into four distinct crossing modes with separate predictability properties.

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

The paper establishes that the first time a càdlàg adapted process crosses a continuous barrier admits a canonical fourfold decomposition into continuous contact, left contact followed by an upward jump, exact hit by jump, and strict overshoot by jump. This decomposition matters because the four modes carry different predictability properties, making the split more useful for random-time analysis than the classical contact-versus-overshoot division. The left-contact component is always accessible and becomes predictable precisely when a no-premature-left-contact condition holds, which the paper shows is necessary and sufficient for the standard running-supremum announcing sequence. When predictable gap crossings are structurally excluded, the corresponding restricted time is totally inaccessible. In the semimartingale setting the decomposition yields sharp compensator criteria and explicit formulas, which are then applied to obtain a boundary-value problem and closed representations for the overshoot mode of a mean-reverting affine jump-diffusion.

Core claim

We show that t admits a canonical fourfold pathwise decomposition into continuous contact, contact from the left followed by an upward jump, exact hit by jump, and strict overshoot by jump from below. This refinement is more informative than the classical contact-versus-overshoot dichotomy for random-time purposes, because it separates modes with different predictability properties. In particular, the left-contact component always defines an accessible stopping time and becomes predictable under a no-premature-left-contact condition, which we prove to be both sufficient and necessary for the canonical running-supremum announcing sequence to work. On the gap side, under a structural exclusion

What carries the argument

The canonical fourfold pathwise decomposition of the first-passage time t into continuous contact, left-contact followed by upward jump, exact jump hit, and strict overshoot by jump.

If this is right

  • The left-contact component defines an accessible stopping time.
  • The no-premature-left-contact condition is necessary and sufficient for the running-supremum announcing sequence to predict the left-contact time.
  • Under structural exclusion of predictable gap-crossings the restricted gap-crossing time is totally inaccessible.
  • Explicit compensator formulas exist for each jump-driven crossing mode and the default-indicator compensator splits into a predictable jump part and a continuous part.
  • For the mean-reverting affine jump-diffusion the overshoot mode satisfies a third-order ODE boundary-value problem whose solution yields a Green-Volterra representation and closed formulas for overshoot and creeping probabilities.

Where Pith is reading between the lines

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

  • The four-mode classification could be used to construct mode-specific Monte Carlo estimators that sample continuous contacts separately from overshoots.
  • Similar pathwise splits may extend to first-exit problems for processes with jumps in multiple dimensions.
  • The compensator decomposition supplies a natural intensity model for point processes that track only certain crossing types.
  • The small-q expansion derived for the discounted problem may admit direct analogues for other parameter regimes in affine models.

Load-bearing premise

The process must have càdlàg paths and the barrier must be continuous so that every crossing can be unambiguously classified into one of the four modes without simultaneous-event ambiguity.

What would settle it

An explicit càdlàg adapted process and continuous barrier in which a left-contact crossing occurs yet fails to be accessible as a stopping time, or in which the running-supremum sequence fails to announce it under the stated no-premature condition.

read the original abstract

Let t be the first-passage time of a continuous barrier by a c{\`a}dl{\`a}g adapted process. We show that t admits a canonical fourfold pathwise decomposition into continuous contact, contact from the left followed by an upward jump, exact hit by jump, and strict overshoot by jump from below. This refinement is more informative than the classical contact-versus-overshoot dichotomy for random-time purposes, because it separates modes with different predictability properties. In particular, the left-contact component always defines an accessible stopping time and becomes predictable under a no-premature-left-contact condition, which we prove to be both sufficient and necessary for the canonical running-supremum announcing sequence to work. On the gap side, under a structural exclusion of predictable gap-crossings, the corresponding restricted time is totally inaccessible. In the semimartingale setting, we obtain a sharp compensator criterion for the predictable-side condition, explicit compensator formulas for the jump-driven crossing modes, and a decomposition of the compensator of the default indicator into its predictable jump part and continuous part. As an application, for a mean-reverting affine jump-diffusion with upward exponential jumps, we derive the boundary-value problem governing the overshoot mode, prove that the differentiated third-order ODE is equivalent to the original problem only when a boundary compatibility condition is retained, and establish verification and uniqueness for the discounted problem. This yields an explicit Green-Volterra representation, a first-order small-q expansion expansion, and, in the undiscounted case, closed formulas for the overshoot and creeping probabilities

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.

Referee Report

0 major / 2 minor

Summary. The paper claims that the first-passage time t through a continuous barrier by a càdlàg adapted process admits a canonical fourfold pathwise decomposition into continuous contact, left-contact followed by an upward jump, exact hit by jump, and strict overshoot by jump from below. This refines the classical contact-versus-overshoot dichotomy by separating modes with distinct predictability properties. The no-premature-left-contact condition is shown to be necessary and sufficient for the left-contact component to be predictable via the canonical running-supremum announcing sequence. Under structural exclusion of predictable gap-crossings the restricted time is totally inaccessible. In the semimartingale setting explicit compensator formulas are derived for the jump-driven modes and the default indicator is decomposed into predictable jump and continuous parts. The application to a mean-reverting affine jump-diffusion with upward exponential jumps derives the governing boundary-value problem for the overshoot mode, proves that the differentiated third-order ODE is equivalent to the original problem only when the boundary compatibility condition is retained, and establishes verification and uniqueness, yielding an explicit Green-Volterra representation, a first-order small-q expansion, and closed formulas for overshoot and creeping probabilities.

Significance. If the central claims hold, the work supplies a useful refinement of first-passage theory that isolates predictability classes and yields explicit compensators, which are directly applicable to default modeling and barrier problems. The pathwise definitions, necessity proofs via counter-examples, and the careful ODE verification in the affine jump-diffusion case (including retention of the compatibility condition) constitute concrete strengths. The resulting closed-form expressions and expansions enhance practical utility without introducing free parameters or circular definitions.

minor comments (2)
  1. [Abstract] Abstract, final sentence: the phrase 'small-q expansion expansion' contains a duplicated word and should be corrected.
  2. [Application] Application section: the boundary compatibility condition is stated as necessary for ODE equivalence, but its explicit functional form could be displayed more prominently (e.g., as a displayed equation) to facilitate implementation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and recommendation of minor revision. We appreciate the recognition of the pathwise decomposition's refinement of predictability classes, the necessity proofs, the compensator formulas, and the detailed verification for the affine jump-diffusion boundary-value problem.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The manuscript defines the fourfold pathwise decomposition of the first-passage time explicitly via cadlag path properties (continuous contact, left-contact-plus-jump, exact jump hit, strict overshoot). Predictability and compensator formulas follow from standard semimartingale announcing-sequence arguments and direct verification of the no-premature-left-contact condition, with necessity proved by counter-example. The affine-jump-diffusion application retains the boundary-compatibility condition when differentiating the ODE and proves uniqueness by standard verification arguments. No equation reduces to a fitted parameter renamed as prediction, no load-bearing premise rests on a self-citation chain, and no ansatz is smuggled via prior work. All steps are independent of the target claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard assumptions of stochastic process theory (cadlag adapted processes, semimartingales) without introducing new free parameters or postulated entities; the boundary compatibility condition is derived rather than fitted.

axioms (2)
  • domain assumption The underlying process is cadlag and adapted to a filtration
    Invoked throughout the abstract as the setting for the first-passage time t.
  • standard math Semimartingale decomposition exists and compensators are well-defined
    Used to obtain explicit compensator formulas for the jump-driven crossing modes.

pith-pipeline@v0.9.0 · 5587 in / 1461 out tokens · 55739 ms · 2026-05-13T18:41:06.702334+00:00 · methodology

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