pith. sign in

arxiv: 2606.04245 · v2 · pith:MOJT3WGDnew · submitted 2026-06-02 · 🧮 math.PR

Properties of a Special Type of Filtration and its Martingale Criteria

Assylliya K. Zhunussova (Lomonosov Moscow State University , Moscow , Russia) This is my paper

Pith reviewed 2026-07-01 07:43 UTC · model grok-4.3

classification 🧮 math.PR
keywords single-jump filtrationoptional projectionstopping timesmartingale criterialocal martingalesadapted processesinitial sigma-algebrameasurability criteria
0
0 comments X

The pith

Enlarging a single-jump filtration by a non-trivial initial sigma-algebra preserves characterizations of stopping times and martingales via optional projections.

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

The paper extends the generalized single-jump filtration of Gushchin by adjoining a non-trivial initial sigma-algebra H. It applies optional projection techniques from the general theory of processes to obtain measurability criteria for random variables together with complete descriptions of stopping times and adapted processes. It further supplies necessary and sufficient conditions that ensure a process remains a martingale or local martingale after the enlargement. A reader would care because these explicit criteria let one verify the martingale property directly when pre-jump information is already present in the model.

Core claim

By leveraging the general theory of processes and optional projection techniques, the authors establish fundamental measurability criteria for random variables and a complete characterization of stopping times and adapted processes in the extended filtration. They derive comprehensive martingale and local martingale criteria, providing necessary and sufficient conditions for the preservation of the martingale property.

What carries the argument

The generalized single-jump filtration enlarged by a non-trivial initial sigma-algebra H, with optional projection techniques supplying the explicit criteria for stopping times, adaptedness, and martingale preservation.

If this is right

  • Stopping times relative to the enlarged filtration admit an explicit description in terms of the jump time and sets from the initial sigma-algebra.
  • A process is adapted to the enlarged filtration if and only if it satisfies a measurability condition expressible via optional projections.
  • A process remains a martingale after enlargement precisely when its compensator satisfies the derived projection identity.
  • The same projection identity supplies necessary and sufficient conditions for the process to remain a local martingale.

Where Pith is reading between the lines

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

  • The same enlargement technique could be tested on other filtrations with a single distinguished jump time to see whether the optional-projection method still yields closed-form criteria.
  • Models that already encode market information before a default time could now use these criteria to check the martingale property without reverting to the original smaller filtration.
  • The approach suggests a route to enlarge other minimal filtrations while retaining tractable martingale tests, provided the optional-projection identities survive the enlargement.

Load-bearing premise

The structural properties of the original generalized single-jump filtration continue to hold after the initial sigma-algebra is adjoined, without fresh derivation of the optional-projection identities.

What would settle it

An explicit counter-example in which adjoining the initial sigma-algebra H causes the optional-projection identities to fail, so that the claimed stopping-time and martingale characterizations no longer hold.

read the original abstract

This article investigates the structural properties of stochastic processes relative to a generalized single jump filtration, extending the framework introduced by A.A. Gushchin (2020) to the case of a non-trivial initial $\sigma$-algebra $\mathscr{H}$. By leveraging the general theory of processes and optional projection techniques, we establish fundamental measurability criteria for random variables and a complete characterization of stopping times and adapted processes. Furthermore, we derive comprehensive martingale and local martingale criteria, providing necessary and sufficient conditions for the preservation of the martingale property in this extended setting.

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

1 major / 2 minor

Summary. The manuscript extends the generalized single-jump filtration framework of Gushchin (2020) by adjoining a non-trivial initial σ-algebra H. Leveraging the general theory of processes and optional-projection techniques, it claims to derive measurability criteria for random variables, complete characterizations of stopping times and adapted processes, and necessary-and-sufficient conditions for a process to remain a martingale or local martingale in the enlarged filtration.

Significance. If the optional-projection identities are shown to be invariant under the enlargement by H, the results would supply a usable extension for modeling jump processes that carry non-trivial initial information. The contribution is incremental, resting directly on the 2020 framework, but could be of practical value in applications such as credit-risk modeling or reliability where initial σ-algebras appear.

major comments (1)
  1. [Abstract and §2 (framework extension)] The central claims rest on the assertion that optional-projection identities from Gushchin (2020) continue to supply necessary-and-sufficient martingale criteria after adjoining H. Because the enlargement modifies the filtration at t=0, the optional σ-algebra, predictable σ-algebra, and explicit form of the optional projection may change; the manuscript invokes these identities without an explicit re-derivation or invariance argument. This is load-bearing for the necessity-and-sufficiency statements in the martingale criteria.
minor comments (2)
  1. [§1] Notation for the enlarged filtration (e.g., the precise definition of the optional σ-algebra generated by the H-augmented processes) should be stated explicitly rather than left implicit by reference to Gushchin (2020).
  2. [Abstract] The abstract states that the criteria are 'comprehensive' and 'necessary and sufficient'; the introduction should clarify whether these hold pathwise or in expectation, and whether they require any integrability assumptions beyond those in the 2020 paper.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for an explicit invariance argument. We agree that the load-bearing nature of the optional-projection identities warrants a dedicated clarification and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and §2 (framework extension)] The central claims rest on the assertion that optional-projection identities from Gushchin (2020) continue to supply necessary-and-sufficient martingale criteria after adjoining H. Because the enlargement modifies the filtration at t=0, the optional σ-algebra, predictable σ-algebra, and explicit form of the optional projection may change; the manuscript invokes these identities without an explicit re-derivation or invariance argument. This is load-bearing for the necessity-and-sufficiency statements in the martingale criteria.

    Authors: We acknowledge the referee's observation. Although the general theory of processes guarantees that optional projections exist with respect to the enlarged filtration, and the single-jump structure for t>0 is unchanged by the initial σ-algebra H, an explicit invariance argument is indeed desirable for rigor. In the revised version we will insert a short subsection in §2 that (i) recalls the definition of the optional projection relative to the enlarged filtration, (ii) shows that the optional and predictable σ-algebras coincide with those of the original Gushchin filtration on the relevant sets, and (iii) verifies that the martingale criteria therefore transfer verbatim. This addition will make the necessity-and-sufficiency statements fully self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: extension applies general theory to cited external framework without self-referential reduction.

full rationale

The paper explicitly positions itself as an extension of Gushchin (2020) by a different author and states that its measurability, stopping-time, and martingale criteria follow from the general theory of processes plus optional projection techniques. No equations or claims in the provided abstract or description reduce any derived result to a fitted parameter, self-definition, or load-bearing self-citation chain; the cited framework is external and the derivation is presented as building upon it rather than presupposing its own outputs. This is the normal, non-circular case of citing prior independent work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the general theory of processes and optional projections as background. No free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • standard math The optional projection of an integrable process onto the filtration exists and satisfies the usual properties.
    Invoked to obtain measurability criteria; standard in the general theory of processes.
  • domain assumption The structural properties of the single-jump filtration from Gushchin (2020) remain valid after adjoining a non-trivial initial sigma-algebra H.
    This preservation is presupposed for the extension to work; it is not re-proved in the abstract.

pith-pipeline@v0.9.1-grok · 5624 in / 1463 out tokens · 21997 ms · 2026-07-01T07:43:02.033970+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Explicit Predictable Compensators for Single Jump Processes with Initial Information

    math.PR 2026-06 unverdicted novelty 5.0

    Derives explicit predictable compensators for cadlag finite-variation processes in single-jump filtrations with initial information via sigma-martingales when integrability fails.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · cited by 1 Pith paper

  1. [1]

    In: Séminaire de Probabilités, IX

    Chou, C.-S., Meyer, P.-A.: Sur la représentation des martingales comme intégrales stochastiques dans les processus ponctuels. In: Séminaire de Probabilités, IX. Lecture Notes in Math., vol. 465, pp. 226-236. Springer (1975). doi:10.1007/BFb0102993. MR436310 https://mathscinet.ams.org/mathscinet/relay-station?mr=436310

    work page doi:10.1007/bfb0102993 1975

  2. [2]

    In: Séminaire de probabilités, IV

    Dellacherie, C.: Un exemple de la théorie générale des processus. In: Séminaire de probabilités, IV. Lecture Notes in Math., vol. 124, pp. 60–70. Springer, Berlin (1970). doi:10.1007/BFb0059333. MR263157 http://www.ams.org/mathscinet-getitem?mr=263157

    work page doi:10.1007/bfb0059333 1970

  3. [3]

    North-Holland Mathematics Studies, vol

    Dellacherie, C., Meyer, P.-A: Probabilities and Potential B: Theory of Discrete and Continuous Parameter Martingales. North-Holland Mathematics Studies, vol. 72, Amsterdam (1982). doi:10.1016/S0304-0208(08)73753-8. MR745449 https://mathscinet.ams.org/mathscinet/relay-station?mr=745449

    work page doi:10.1016/s0304-0208(08)73753-8 1982

  4. [4]

    Gushchin, A.A.: Single jump filtrations and local martingales. Mod. Stoch. Theory Appl. 7(2), 135–156 (2020). doi:10.15559/20-VMSTA153. MR4120612 https://mathscinet.ams.org/mathscinet/relay-station?mr=4120612

    work page doi:10.15559/20-vmsta153 2020

  5. [5]

    (2021): Single Jump Filtrations: Preservation of the Local Martingale Property with Respect to the Filtration Generated by the Local Martingale

    Gushchin, A.A., Zhunussova, A.K. (2021): Single Jump Filtrations: Preservation of the Local Martingale Property with Respect to the Filtration Generated by the Local Martingale. In: Karapetyants, A.N., Pavlov, I.V., Shiryaev, A.N. (eds) Operator Theory and Harmonic Analysis. OTHA 2020. Springer Proceedings in Mathematics & Statistics, vol 358. Springer, C...

    work page doi:10.1007/978-3-030-76829-4 2021

  6. [6]

    Stochastic Processes and Their Applications 126(2), 337-359 (2016)

    Herdegen, M., Herrmann, S.: Single jump processes and strict local martingales. Stochastic Processes and Their Applications 126(2), 337-359 (2016). doi:10.1016/j.spa.2015.09.003. MR3434986 https://mathscinet.ams.org/mathscinet/relay-station?mr=3434986

    work page doi:10.1016/j.spa.2015.09.003 2016

  7. [7]

    Jacod, J., Shiryaev, A. N. (2003): Limit Theorems for Stochastic Processes (2nd ed.). Grundlehren der mathematischen Wissenschaften, Vol. 288. Springer Berlin, Heidelberg. doi:10.1007/978-3-662-05265-5. MR1943877 https://mathscinet.ams.org/mathscinet/relay-station?mr=1943877

    work page doi:10.1007/978-3-662-05265-5 2003

  8. [8]

    Applications of Mathematics, vol

    Protter, P.E.(2005): Stochastic Integration and Differential Equations. Applications of Mathematics, vol. 21, Springer-Verlag, Berlin, Heidelberg. doi:10.1007/978-3-662-10061-5. MR2273672 https://mathscinet.ams.org/mathscinet/relay-station?mr=2273672

    work page doi:10.1007/978-3-662-10061-5 2005

  9. [9]

    On a question by Alexander Cherny

    Ruf, J.: The uniform integrability of martingales. On a question by Alexander Cherny. Stochastic Processes and Their Applications 125(10), 3657–3662 (2015). doi:10.1016/j.spa.2015.04.002

    work page doi:10.1016/j.spa.2015.04.002 2015

  10. [10]

    N.(1996): Probability (2nd ed.)

    Shiryaev, A. N.(1996): Probability (2nd ed.). Graduate Texts in Mathematics, Vol. 95. Springer New York. doi:10.1007/978-1-4757-2539-1. MR1368405 https://mathscinet.ams.org/mathscinet/relay-station?mr=1368405

    work page doi:10.1007/978-1-4757-2539-1 1996