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arxiv: 2606.20925 · v1 · pith:D7AFJKAOnew · submitted 2026-06-18 · 🧮 math.PR

Explicit Predictable Compensators for Single Jump Processes with Initial Information

Assylliya K. Zhunussova This is my paper

Pith reviewed 2026-06-26 15:45 UTC · model grok-4.3

classification 🧮 math.PR
keywords single jump filtrationpredictable compensatorsigma-martingaleinitial informationheavy-tailed jumpscadlag processfinite variationDoob-Meyer decomposition
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The pith

Explicit compensators for single-jump processes exist via σ-martingales even without integrability.

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

The paper develops an explicit way to find predictable compensators for adapted cadlag processes of finite variation in a single-jump filtration that includes initial information. Classical decompositions fail for heavy-tailed jumps lacking integrability, so the authors turn to σ-martingale theory. They give necessary and sufficient conditions for a process to be a σ-martingale and compute the compensator of a weighted process explicitly. This produces a clear link between the continuous drift term and the expected size of the jump. Readers interested in stochastic processes with limited regularity would find this useful for handling cases where standard tools break down.

Core claim

In the single jump filtration augmented with a sub-sigma-algebra H, for an adapted cadlag finite variation process, the predictable compensator can be constructed explicitly by applying σ-martingale theory to a suitably weighted version of the process, establishing necessary and sufficient conditions for σ-martingale property and yielding an explicit relation between the continuous drift and the expected jump component.

What carries the argument

The σ-martingale decomposition applied to the weighted process in the single-jump filtration with initial information H, which bypasses the integrability requirement of the classical Doob-Meyer theorem.

If this is right

  • The compensator is given by an explicit formula involving the drift and jump expectation.
  • Processes that satisfy the σ-martingale conditions have their jumps compensated in a predictable way.
  • The approach works even when the jump size has a heavy tail and is not integrable.
  • Initial information H is incorporated directly into the filtration without changing the explicit form.

Where Pith is reading between the lines

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

  • This could extend to other jump processes if they can be reduced to single jumps.
  • Applications in mathematical finance might include models with infinite moments.
  • Further work could derive similar explicit forms for the compensator in multi-jump settings.

Load-bearing premise

The single jump filtration with added initial information allows σ-martingale theory to apply where classical martingale decompositions cannot due to lack of integrability.

What would settle it

Observe a specific heavy-tailed single-jump process where the constructed compensator does not turn the process into a σ-martingale, or where the relation between drift and jump expectation fails to hold.

read the original abstract

We study the predictable compensators of stochastic processes in a single jump filtration augmented with initial information represented by a sub-$\sigma$-algebra $\mathcal{H}$. We consider adapted c\`adl\`ag processes of finite variation and give an explicit construction of their predictable compensators. The main difficulty arises when the jump size has a heavy tail and lacks integrability, so that the classical Doob-Meyer decomposition does not apply. To overcome this, we use the theory of $\sigma$-martingales. We establish necessary and sufficient conditions for a process to be a $\sigma$-martingale and explicitly compute the compensator of a suitably weighted process. This yields an explicit relation between the continuous drift and the expected jump component of the original process.

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 / 3 minor

Summary. The manuscript develops explicit constructions of the predictable compensators for adapted càdlàg processes of finite variation in a single-jump filtration augmented by an initial sub-σ-algebra ℋ. When the jump size lacks integrability, classical Doob-Meyer fails, so the authors invoke σ-martingale theory to obtain necessary and sufficient conditions for a process to be a σ-martingale and to compute the compensator of a suitably weighted version of the process; this produces an explicit relation between the continuous drift and the expected jump component.

Significance. If the explicit formulas and conditions are correctly derived, the work supplies a concrete extension of decomposition results to heavy-tailed single-jump settings with initial information, a setting that arises in applications where integrability assumptions are unrealistic. The parameter-free character of the resulting drift-jump relation and the use of established σ-martingale machinery are strengths.

minor comments (3)
  1. [Section 2] §2 (or wherever the single-jump filtration is defined): the notation for the augmented filtration (ℱ_t = ℋ ∨ σ({τ ≤ s, X_s : s ≤ t})) should be stated once explicitly to avoid ambiguity when the compensator formulas are written later.
  2. [Main theorem] The statement of the main theorem (presumably Theorem 3.1 or 4.2) would benefit from a short remark clarifying whether the weighting function used to restore integrability is chosen independently of the law of the jump or is allowed to depend on it.
  3. [Introduction or §3] A brief comparison paragraph with the classical compensator formula (when E[|ΔX|] < ∞) would help readers see precisely where the σ-martingale correction appears.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript, the recognition of its significance in extending decomposition results to heavy-tailed single-jump settings, and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation applies established theory

full rationale

The paper derives explicit compensators for single-jump processes by applying standard σ-martingale theory to the augmented filtration, establishing necessary and sufficient conditions for σ-martingales and relating drift to jump components when integrability fails for Doob-Meyer. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear; the central claims rest on external σ-martingale results applied to the specific setting rather than reducing to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no specific free parameters, axioms, or invented entities can be identified from the provided information.

pith-pipeline@v0.9.1-grok · 5648 in / 1063 out tokens · 27132 ms · 2026-06-26T15:45:14.944232+00:00 · methodology

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Reference graph

Works this paper leans on

8 extracted references · 6 canonical work pages

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