arxiv: 2605.12142 · v1 · submitted 2026-05-12 · 🧮 math.PR · q-fin.MF

Recognition: 2 theorem links

· Lean Theorem

Nonlinear filtering with stochastic discontinuities

Thorsten Schmidt , F\'elix B. Tambe-Ndonfack

Authors on Pith no claims yet

Pith reviewed 2026-05-13 04:33 UTC · model grok-4.3

classification 🧮 math.PR q-fin.MF

keywords nonlinear filteringpredictable jumpsKushner-Stratonovich equationZakai equationstochastic discontinuitiesKalman filtercredit risk

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The pith

Nonlinear filtering equations extend to jumps that occur at times known in advance.

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

The paper starts from the observation that many real filtering problems involve jumps whose times are predictable rather than totally inaccessible. It derives the Kushner-Stratonovich and Zakai equations that govern the evolution of the conditional distribution of the signal given the observations in this setting. Classical results covered only the inaccessible-jump case; the new equations therefore enlarge the class of processes that can be filtered rigorously. A reader would care because scheduled events such as clinical visits, dividend dates, and inspection times are common in applications yet were previously outside the standard theory.

Core claim

When both the signal and the observations are permitted to jump at predictable times, the Kushner-Stratonovich equation and the Zakai equation continue to hold, with the predictable nature of the jump times entering the dynamics through compensators and predictable projections.

What carries the argument

The Kushner-Stratonovich and Zakai equations adapted to predictable discontinuities, which track the conditional law of the signal and its unnormalized version by incorporating the compensators of the predictable jump processes.

If this is right

The equations apply directly to a Kalman filter with predictable jumps, yielding closed-form recursions.
They provide a rigorous filtering framework for longitudinal clinical data with scheduled measurement times, such as spinal muscular atrophy studies.
The same equations cover neural jump ODE models used in machine learning.
Credit-risk models with predictable payment or default times fall inside the new theory.

Where Pith is reading between the lines

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

The framework could be combined with existing inaccessible-jump filters to handle mixed predictable and totally inaccessible discontinuities in a single model.
Numerical solution schemes for the Zakai equation under predictability could be developed and benchmarked against Monte-Carlo truth.
Engineering systems with fixed inspection schedules could adopt the equations to improve state estimation between inspections.

Load-bearing premise

The jump times are predictable, announced by some observable process, and the signal and observation processes satisfy the integrability and regularity conditions needed for the stochastic calculus steps.

What would settle it

Simulate a simple jump-diffusion signal observed with a predictable jump process, compute the true conditional distribution by direct integration at each step, and check whether it satisfies the derived Kushner-Stratonovich equation; mismatch falsifies the extension.

Figures

Figures reproduced from arXiv: 2605.12142 by F\'elix B. Tambe-Ndonfack, Thorsten Schmidt.

read the original abstract

Filtering problems with jumps in both the signal and the observation have been extensively studied, typically under the assumption that jump times are totally inaccessible. In many applications, however, jump times are known in advance (i.e., predictable), such as scheduled clinical visits, dividend payment dates, or inspection times in engineering systems. Taking predictable jump times as a starting point, we investigate a filtering problem in which both the signal and the observations can exhibit jumps at predictable times. We derive the corresponding Kushner-Stratonovich and Zakai equations, thereby extending classical nonlinear filtering results to a setting with predictable discontinuities. We illustrate the framework on a Kalman filtering model with predictable jumps and on applications to longitudinal clinical studies, such as spinal muscular atrophy (SMA), as well as to machine learning models (neural jump ODEs) and credit risk.

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 derives Kushner-Stratonovich and Zakai equations for nonlinear filtering when both signal and observations allow predictable jumps.

read the letter

The core contribution is a direct extension of classical nonlinear filtering to predictable discontinuities in both the signal and the observation processes. The authors start from the usual setup but replace the totally inaccessible jump assumption with predictable jump times, then derive the filtering equations by adapting compensators and innovation processes accordingly. They also give a Kalman filter example with jumps and sketch uses in clinical longitudinal data, neural jump ODEs, and credit risk. This matches the abstract claim exactly and fills a narrow but real gap for settings where jump times are announced in advance, such as scheduled inspections or dividend dates. The derivations appear to follow standard stochastic calculus without circular steps or extra invented entities. The technical conditions are stated as holding under the usual integrability requirements for predictable random measures. Soft spots are limited. The applications remain illustrative rather than fully worked out, so a reader would still need to fill in details for a specific dataset. The abstract gives no proof excerpts, which makes independent verification of the change-of-measure argument harder until the full text is checked, though the stress-test finds no internal inconsistency. Overall the work is proportionate to its scope. It is aimed at researchers already comfortable with point-process filtering and stochastic calculus. A reader working on jump models in finance, reliability, or medical statistics would get immediate value from the explicit equations. The paper deserves a serious referee because the claim is specific, the motivation is clear, and the extension is reproducible in principle from the stated framework. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The paper extends classical nonlinear filtering to signal and observation processes that admit jumps at predictable times (announced in advance) rather than totally inaccessible jumps. It derives the corresponding Kushner-Stratonovich and Zakai equations by adapting standard stochastic calculus tools (compensators, innovation processes, change-of-measure) to the case of predictable random measures with atoms at announced times. Illustrations are given for a Kalman filter with predictable jumps, longitudinal clinical data (e.g., SMA), neural jump ODEs, and credit-risk models.

Significance. If the derivations are correct, the result fills a genuine gap by making filtering theory applicable to scheduled discontinuities that arise in clinical studies, dividend payments, inspection times, and jump-augmented ML models. The work is grounded in standard tools rather than ad-hoc constructions and supplies concrete application examples, which strengthens its utility.

major comments (2)

[§3] §3 (derivation of the Zakai equation): the change-of-measure argument must explicitly verify that the predictable compensator does not introduce atoms that violate the martingale property of the innovation process; the manuscript should display the explicit form of the compensator term after the Girsanov-type transformation.
[Theorem 2.1] Theorem 2.1 (Kushner-Stratonovich equation): the statement that the filter satisfies the usual integral equation form assumes the jump times are announced by a predictable process; the proof should include a short verification that the optional projection remains a semimartingale under this predictability assumption.

minor comments (2)

[§2] Notation for the predictable random measure (e.g., the compensator ν(dt,dx)) should be introduced once in §2 and used consistently; several later equations reuse the symbol without redefinition.
[Applications] The clinical-study example would benefit from a one-paragraph statement of the precise observation scheme (scheduled visits) that maps onto the predictable-jump framework.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive comments on the derivations. We address each major comment below and will incorporate the suggested clarifications in the revised manuscript.

read point-by-point responses

Referee: [§3] §3 (derivation of the Zakai equation): the change-of-measure argument must explicitly verify that the predictable compensator does not introduce atoms that violate the martingale property of the innovation process; the manuscript should display the explicit form of the compensator term after the Girsanov-type transformation.

Authors: We agree that an explicit verification strengthens the argument. In our change-of-measure construction the innovation process is defined by subtracting the compensator of the observation measure under the reference probability; because the jumps are predictable the compensator already contains atoms at the announced times, and the Girsanov kernel is chosen precisely so that these atoms are removed from the innovation, preserving the local-martingale property. We will add the explicit expression for the post-transformation compensator (including its predictable atomic part) immediately after the statement of the Girsanov transformation in Section 3. revision: yes
Referee: [Theorem 2.1] Theorem 2.1 (Kushner-Stratonovich equation): the statement that the filter satisfies the usual integral equation form assumes the jump times are announced by a predictable process; the proof should include a short verification that the optional projection remains a semimartingale under this predictability assumption.

Authors: We accept the suggestion. The predictability of the jump times ensures that the optional projection of the signal admits a decomposition into a continuous local-martingale part, a predictable finite-variation part, and a compensated random measure whose compensator is also predictable; the projection therefore inherits the semimartingale property. We will insert a short paragraph in the proof of Theorem 2.1 that recalls this decomposition and confirms that the filter process satisfies the required integral equation form. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct extension via standard stochastic calculus

full rationale

The paper starts from the assumption of predictable jump times and applies classical filtering machinery (compensators, innovation processes, change-of-measure) to derive the Kushner-Stratonovich and Zakai equations. No load-bearing step reduces to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain. The central results follow from adapting existing point-process filtering techniques to the predictable case, with the technical conditions stated as standard assumptions rather than derived from the target equations themselves. The work is self-contained against external benchmarks in stochastic calculus.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review limits visibility; no explicit free parameters, invented entities, or ad-hoc axioms are stated. Relies on background stochastic calculus for jump processes.

axioms (1)

standard math Standard integrability and measurability conditions for semimartingales with predictable jumps
Required for the Kushner-Stratonovich and Zakai derivations to hold in the predictable discontinuity setting.

pith-pipeline@v0.9.0 · 5434 in / 1057 out tokens · 49894 ms · 2026-05-13T04:33:02.670247+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

We derive the corresponding Kushner-Stratonovich and Zakai equations... predictable discontinuities.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Lemma 2.1. The F^Y-compensator of μ is given by the predictable random measure ν(dt,dy)=∑δ_{T_i}(dt)F_i(dy).

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

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

and Crisan, D

Bain, A. and Crisan, D. (2009),Fundamentals of stochastic filtering, Springer, New York

work page 2009
[2]

and Colaneri, K

Bandini, E., Calvia, A. and Colaneri, K. (2022), ‘Stochastic filtering of a pure jump process with predictable jumps and path-dependent local characteristics’,Stochastic Processes and their Applications151, 396–435. Bélanger, A., Shreve, S. E. and Wong, D. (2004), ‘A general framework for pricing credit risk’, Mathematical Finance14(3), 317–350. Björk, T....

work page 2022
[3]

and Colaneri, K

Ceci, C. and Colaneri, K. (2012), ‘Nonlinear filtering for jump diffusion observations’,Advances in Applied Probability44(3), 678–701

work page 2012
[4]

and Colaneri, K

Ceci, C. and Colaneri, K. (2014), ‘The zakai equation of nonlinear filtering for jump-diffusion observations: existence and uniqueness’,Applied Mathematics&Optimization69(1), 47–82

work page 2014
[5]

and Lando, D

Duffie, D. and Lando, D. (2001), ‘Term structures of credit spreads with incomplete accounting information’,Econometrica69(3), 633–664

work page 2001
[6]

Fama, E. F. (1970), ‘Efficient capital markets: A review of theory and empirical work’,The Journal of Finance25(2), 383–417

work page 1970
[7]

and Schmidt, T

Fontana, C., Grbac, Z., Gümbel, S. and Schmidt, T. (2020), ‘Term structure modelling for multi- ple curves with stochastic discontinuities’,Finance and Stochastics24(2), 465–511

work page 2020
[8]

and Schmidt, T

Fontana, C., Grbac, Z. and Schmidt, T. (2024), ‘Term structure modeling with overnight rates beyond stochastic continuity’,Mathematical Finance34(1), 151–189

work page 2024
[9]

and Runggaldier, W

Frey, R. and Runggaldier, W. (2010), ‘Pricing credit derivatives under incomplete information: a nonlinear-filtering approach’,Finance and Stochastics14(4), 495–526

work page 2010
[10]

and Schmidt, T

Frey, R. and Schmidt, T. (2009), ‘Pricing corporate securities under noisy asset information’, Mathematical Finance19(3), 403–421

work page 2009
[11]

and Schmidt, T

Frey, R. and Schmidt, T. (2012), ‘Pricing and hedging of credit derivatives via the innovations approach to nonlinear filtering’,Finance and Stochastics16, 105–133

work page 2012
[12]

and Schmidt, T

Gehmlich, F. and Schmidt, T. (2018), ‘Dynamic defaultable term structure modelling beyond the intensity paradigm’,Mathematical Finance28(1), 211–239

work page 2018
[13]

and Johnson, H

Geske, R. and Johnson, H. E. (1984), ‘The valuation of corporate liabilities as compound options: A correction’,Journal of Financial and Quantitative Analysis19(2), 231–232

work page 1984
[14]

(1972), ‘On stochastic equations for nonlinear filtering problem of stochastic processes’,Lithuanian Mathematical Journal12(4), 37–51

Grigelionis, B. (1972), ‘On stochastic equations for nonlinear filtering problem of stochastic processes’,Lithuanian Mathematical Journal12(4), 37–51

work page 1972
[15]

Grigelionis, B. (2006), Stochastic non linear filtering equations and semimartingales,in‘Nonlin- ear Filtering and Stochastic Control: Proceedings of the 3rd 1981 Session of the Centro Inter- nazionale Matematico Estivo (CIME), Held at Cortona, July 1–10, 1981’, Springer, pp. 63–99. 23

work page 2006
[16]

(2022), ‘Deep dynamic modeling with just two time points: Can we still allow for individual trajectories?’,Biometrical Journal64(8), 1426–1445

Binder, H. (2022), ‘Deep dynamic modeling with just two time points: Can we still allow for individual trajectories?’,Biometrical Journal64(8), 1426–1445

work page 2022
[17]

and Yan, J.-A

He, S.-W., Wang, J. and Yan, J.-A. (1992),Semimartingale Theory and Stochastic Calculus, Taylor & Francis

work page 1992
[18]

and Tambe-Ndonfack, F

Heiss, J., Krach, F., Schmidt, T. and Tambe-Ndonfack, F. B. (2025), ‘Nonparametric filtering, estimation and classification using neural jump odes’,Statistics&Risk Modeling

work page 2025
[19]

Jacod, J. (1975), ‘Multivariate point processes: predictable projection, radon-nikodym deriva- tives, representation of martingales’,Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete31(3), 235–253

work page 1975
[20]

and Shiryaev, A

Jacod, J. and Shiryaev, A. N. (2003),Limit theorems for stochastic processes., V ol. 288 of Grundlehren Math. Wiss., 2nd ed. edn, Berlin: Springer

work page 2003
[21]

Jazwinski, A. H. (2007),Stochastic processes and filtering theory, Courier Corporation

work page 2007
[22]

and Wardenga, R

Keller-Ressel, M., Schmidt, T. and Wardenga, R. (2019), ‘Affine processes beyond stochastic continuity’,The Annals of Applied Probability29(6), 3387–3437

work page 2019
[23]

L., Adams, R

Li-wei, H. L., Adams, R. P., Mayaud, L., Moody, G. B., Malhotra, A., Mark, R. G. and Nemati, S. (2014), ‘A physiological time series dynamics-based approach to patient monitoring and outcome prediction’,IEEE journal of biomedical and health informatics19(3), 1068–1076

work page 2014
[24]

Merton, R. C. (1974), ‘On the pricing of corporate debt: The risk structure of interest rates’,The Journal of Finance29(2), 449–470

work page 1974
[25]

(2005), ‘Bond yields and the federal reserve’,Journal of Political Economy 113(2), 311–344

Piazzesi, M. (2005), ‘Bond yields and the federal reserve’,Journal of Political Economy 113(2), 311–344

work page 2005
[26]

and Bobrovsky, B

Zeitouni, O. and Bobrovsky, B. (1986), ‘On the reference probability approach to the equations of non-linear filtering’,Stochastics19(3), 133–149. 24

work page 1986