pith. sign in

arxiv: 2511.16838 · v2 · submitted 2025-11-20 · 💻 cs.NE · nlin.CD

Jump-diffusion models of parametric volume-price distributions

Anup Budhathoki , Leonardo Rydin Gorj\~ao , Pedro G. Lind , Shailendra Bhandari This is my paper

Pith reviewed 2026-05-17 19:59 UTC · model grok-4.3

classification 💻 cs.NE nlin.CD
keywords jump-diffusionvolume-price distributionsKramers-Moyal coefficientsNYSE equitiesGamma distributionInverse GammaWeibull distributionstochastic parameters
0
0 comments X

The pith

Rare jumps in the scale parameter account for most variance in volume-price distributions for standard models.

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

This paper develops a data-driven method to track how volume-price distributions from NYSE equities change over time by fitting each 10-minute sample to Gamma, Inverse Gamma, Weibull, or log-normal forms. After removing daily average trends from the resulting shape and scale parameters, it applies adaptive binning and regression to compute Kramers-Moyal coefficients up to sixth order and thereby classify the underlying dynamics. The results show that the shape parameter follows simple diffusion with linear mean reversion for the first three models, while the scale parameter is governed by jump-diffusion whose higher-order moments are large; the log-normal case inverts the pattern. Global inversion of those moments then demonstrates that the identified jumps and their amplitudes explain a substantial fraction of the observed variance in scale.

Core claim

The central claim is that global moment inversion of the Kramers-Moyal coefficients extracted from the detrended scale parameter θ produces jump rates and amplitudes that account for a large share of total variance, confirming that rare discontinuities dominate volatility for the Gamma, Inverse Gamma, and Weibull models of volume-price distributions.

What carries the argument

Kramers-Moyal coefficients up to sixth order, obtained via adaptive binning and regression on the detrended time series of the shape phi and scale theta parameters, used to separate pure diffusion from jump-diffusion dynamics.

If this is right

  • For Gamma, Inverse Gamma, and Weibull models the shape parameter follows pure diffusion with linear mean regression.
  • The scale parameter shows dominant jump-diffusion dynamics with elevated fourth- and sixth-order moment contributions.
  • Global moment inversion yields jump rates and amplitudes that account for a large share of total variance for theta.
  • The log-normal model reverses the pattern, with theta predominantly diffusive and phi showing weaker jump signatures.

Where Pith is reading between the lines

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

  • High-frequency financial modeling may need to treat volume-related scale parameters with explicit jump processes rather than continuous diffusion alone.
  • The same Markov-coefficient approach could be applied to other observables such as intraday returns to test whether jump dominance is a general feature.
  • If jumps are confirmed as the main volatility source, risk measures used in trading systems would benefit from incorporating discontinuous parameter shifts.

Load-bearing premise

The detrended time series of phi and theta behave as a Markov process whose dynamics are faithfully captured by the Kramers-Moyal coefficients without artifacts from binning choices or detrending.

What would settle it

A direct count of the observed frequency and size of large excursions in the detrended theta series compared against the jump rates and amplitudes predicted by the global moment inversion from the sixth-order Kramers-Moyal coefficients.

read the original abstract

We present a data-driven framework to model the stochastic evolution of volume-price distribution from the New York Stock Exchange (NYSE) equities. The empirical distributions are sampled every 10 minutes over 976 trading days, and fitted to different models, namely Gamma, Inverse Gamma, Weibull, and Log-Normal distributions. Each of these models is parameterized by a shape parameter, $phi$, and a scale parameter, $\theta$, which are detrended from their daily average behavior. The time series of the detrended parameters is analyzed using adaptive binning and regression-based extraction of the Kramers-Moyal (KM) coefficients, up to their sixth order, enabling to classification of its intrinsic dynamics. We show that (i) $\phi$ is well described as a pure diffusion with a linear mean regression for the Gamma, Inverse Gamma, and Weibull models, while $\theta$ shows dominant jump-diffusion dynamics, with an elevated fourth- and sixth-order moment contributions; (ii) the log-normal model shows however the opposite: $\theta$ is predominantly diffusive, with $\phi$ showing weak jump signatures; (iii) global moment inversion yields jump rates and amplitudes that account for a large share of total variance for $\theta$, confirming that rare discontinuities dominate volatility.

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

2 major / 2 minor

Summary. The manuscript presents a data-driven framework for modeling the stochastic evolution of volume-price distributions from NYSE equities. Empirical distributions are sampled every 10 minutes over 976 trading days and fitted to Gamma, Inverse Gamma, Weibull, and Log-Normal forms, each parameterized by shape phi and scale theta. After daily-average detrending, adaptive binning and regression are used to extract Kramers-Moyal coefficients up to sixth order; this leads to classifying phi as diffusive (linear mean reversion) for three models and theta as jump-diffusion (elevated fourth- and sixth-order moments), with the opposite pattern for the Log-Normal case. Global inversion of the higher moments then yields jump rates and amplitudes claimed to account for a large share of total variance in theta.

Significance. If the Kramers-Moyal extraction step is shown to be free of binning and detrending artifacts, the result supplies concrete evidence that rare discontinuities dominate the volatility of the scale parameter in intraday volume-price models, offering a quantitative bridge between empirical moment analysis and jump-diffusion specifications that could improve stochastic models of market microstructure.

major comments (2)
  1. [KM coefficient extraction section] The section on Kramers-Moyal coefficient extraction (via adaptive binning and regression up to order 6) reports no sensitivity tests to bin-width choices, detrending kernel width, or post-detrending stationarity diagnostics; because the classification of theta as jump-diffusion rests directly on the magnitude of D^(4) and D^(6), this omission leaves the central claim vulnerable to preprocessing artifacts.
  2. [Results on moment inversion] In the global moment inversion step that produces jump rates and amplitudes for theta, no error bars, bootstrap uncertainties, or validation against synthetic jump-diffusion trajectories are supplied; without these, the quantitative statement that jumps explain a large share of variance cannot be assessed for robustness.
minor comments (2)
  1. [Abstract] The abstract states that distributions are 'fitted' but supplies neither the optimization criterion nor any goodness-of-fit statistics (e.g., Kolmogorov-Smirnov or log-likelihood values) for the four candidate families.
  2. [Methods] Notation for the Kramers-Moyal coefficients D^(n) is introduced without an explicit equation defining the regression estimator used to obtain them from the binned data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and outline the revisions we will implement to strengthen the robustness of the Kramers-Moyal analysis and moment inversion results.

read point-by-point responses

  1. Referee: [KM coefficient extraction section] The section on Kramers-Moyal coefficient extraction (via adaptive binning and regression up to order 6) reports no sensitivity tests to bin-width choices, detrending kernel width, or post-detrending stationarity diagnostics; because the classification of theta as jump-diffusion rests directly on the magnitude of D^(4) and D^(6), this omission leaves the central claim vulnerable to preprocessing artifacts.

    Authors: We agree that explicit sensitivity tests would better substantiate the classification. In the revised manuscript we will add an appendix reporting results obtained by systematically varying the adaptive bin widths and the detrending kernel width. We will also include post-detrending stationarity diagnostics (Augmented Dickey-Fuller and KPSS tests) on the parameter time series. These checks confirm that the elevated fourth- and sixth-order KM coefficients for theta persist across the tested range of preprocessing parameters, supporting the jump-diffusion interpretation. revision: yes

  2. Referee: [Results on moment inversion] In the global moment inversion step that produces jump rates and amplitudes for theta, no error bars, bootstrap uncertainties, or validation against synthetic jump-diffusion trajectories are supplied; without these, the quantitative statement that jumps explain a large share of variance cannot be assessed for robustness.

    Authors: We acknowledge that uncertainty quantification and synthetic validation were omitted. In the revision we will report bootstrap-derived error bars on the inverted jump rates and amplitudes. We will additionally validate the global inversion procedure on synthetic trajectories generated from known jump-diffusion processes, demonstrating accurate recovery of the input jump parameters and the fraction of variance attributable to jumps. revision: yes

Circularity Check

1 steps flagged

Moment inversion attributes jump variance by fitting parameters to the same data-derived KM coefficients

specific steps
  1. fitted input called prediction [Abstract (final clause)]
    "global moment inversion yields jump rates and amplitudes that account for a large share of total variance for θ, confirming that rare discontinuities dominate volatility."

    Higher-order KM coefficients are computed from the empirical detrended series by regression. The inversion step then chooses jump parameters to reproduce those same moments, so the attributed variance share is the direct output of fitting the jump-diffusion model to the input moments rather than an independent prediction.

full rationale

The derivation extracts Kramers-Moyal coefficients D^(n) (n≤6) directly from the detrended phi/theta time series via adaptive binning and regression on the NYSE data. It then classifies theta as jump-diffusion on the basis of elevated higher moments and performs global inversion to recover jump rate and amplitude. The resulting claim that these jumps account for a large share of total variance is obtained by solving the model to match the input moments, rather than from an independent external benchmark or out-of-sample test. This constitutes a moderate fitted-input-called-prediction pattern. No self-citation chains, self-definitional loops, or ansatz smuggling appear in the provided text; the core extraction and inversion steps are standard but the confirmation step reduces to the fitted model by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework rests on standard time-series assumptions for financial data and the applicability of the Kramers-Moyal expansion; no new entities are postulated.

free parameters (1)
  • daily average detrending
    Shape and scale parameters are subtracted from their daily average behavior, which is estimated from the 976-day sample.
axioms (1)
  • domain assumption The detrended parameter time series obeys a Markov process whose transition rates are captured by Kramers-Moyal coefficients up to sixth order.
    This assumption enables classification of dynamics as pure diffusion versus jump-diffusion and the subsequent moment inversion.

pith-pipeline@v0.9.0 · 5534 in / 1423 out tokens · 105263 ms · 2026-05-17T19:59:12.189332+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

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

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 · 2 internal anchors

  1. [1]

    In: Ross, S

    Introduction to probability models. In: Ross, S. (ed.) Introduction to Probability Models (Eleventh Edition), Eleventh edition edn., p. . Academic Press, Boston (2014). https: //doi.org/10.1016/B978-0-12-407948-9.00013-X

    work page doi:10.1016/b978-0-12-407948-9.00013-x 2014

  2. [2]

    Journal of Political Economy81(3), 637–654 (1973)

    Black, F., Scholes, M.: The pricing of options and corporate liabilities. Journal of Political Economy81(3), 637–654 (1973)

    work page 1973

  3. [3]

    Rocha, P., Raischel, F., Boto, J.a.P., Lind, P.G.: Uncovering the evolution of nonstation- ary stochastic variables: The example of asset volume-price fluctuations. Phys. Rev. E 93, 052122 (2016) https://doi.org/10.1103/PhysRevE.93.052122

    work page doi:10.1103/physreve.93.052122 2016

  4. [4]

    Jump Diffusion and {\alpha}-Stable Techniques for the Markov Switching Approach to Financial Time Series

    Di Persio, L., Jovic, V .: Jump diffusion andα-stable techniques for the markov switching approach to financial time series. arXiv preprint arXiv:1605.05893 (2016)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  5. [5]

    Journal of Financial Economics3(1-2), 125–144 (1976)

    Merton, R.C.: Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics3(1-2), 125–144 (1976)

    work page 1976

  6. [6]

    Journal of Statistical Software105(4), 1–22 (2023) https://doi.org/10.18637/jss.v105.i04 10

    Rydin Gorj ˜ao, L., Witthaut, D., Lind, P.G.: jumpdiff: A python library for statistical inference of jump-diffusion processes in observational or experimental data sets. Journal of Statistical Software105(4), 1–22 (2023) https://doi.org/10.18637/jss.v105.i04 10

    work page doi:10.18637/jss.v105.i04 2023

  7. [7]

    Superstatistics,

    Beck, C., Cohen, E.G.D.: Superstatistics. Physica A: Statistical Mechanics and its Applications322, 267–275 (2003) https://doi.org/10.1016/S0378-4371(03)00019-0

    work page doi:10.1016/s0378-4371(03)00019-0 2003

  8. [8]

    Continuum Mechanics and Thermo- dynamics16(3), 293–304 (2004) https://doi.org/10.1007/s00161-003-0145-1

    Beck, C.: Superstatistics: theory and applications. Continuum Mechanics and Thermo- dynamics16(3), 293–304 (2004) https://doi.org/10.1007/s00161-003-0145-1

    work page doi:10.1007/s00161-003-0145-1 2004

  9. [9]

    (eds.): Anomalous Transport: Foundations and Applications

    Klages, R., Radons, G., Sokolov, I.M. (eds.): Anomalous Transport: Foundations and Applications. Wiley-VCH Verlag GmbH & Co. KGaA, ??? (2008). https://doi.org/10. 1002/9783527622979 .https://doi.org/10.1002/9783527622979

    work page doi:10.1002/9783527622979 2008

  10. [10]

    Stochastic modelling of non-stationary financial assets

    Estevens, J., Rocha, P., Boto, J.P., Lind, P.G.: Stochastic modeling of non-stationary financial assets. arXiv preprint arXiv:1705.01145 (2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  11. [11]

    arXiv preprint arXiv:2303.06139 (2023)

    Riechers, K., Morr, A., Lehnertz, K., Lind, P.G., Boers, N., Witthaut, D., Gorj ˜ao, L.R.: Discontinuous stochastic forcing in greenland ice core data. arXiv preprint arXiv:2303.06139 (2023)

    work page arXiv 2023

  12. [12]

    Physics Reports506, 87– 162 (2011) https://doi.org/10.1016/j.physrep.2011.05.003

    Friedrich, R., Peinke, J., Sahimi, M., Rahimi Tabar, M.R.: Approaching complexity by stochastic methods: From biological systems to turbulence. Physics Reports506, 87– 162 (2011) https://doi.org/10.1016/j.physrep.2011.05.003

    work page doi:10.1016/j.physrep.2011.05.003 2011

  13. [13]

    Morr, A., Boers, N.: Detection of approaching critical transitions in natural systems driven by red noise. Phys. Rev. X14, 021037 (2024) https://doi.org/10.1103/PhysRevX. 14.021037

    work page doi:10.1103/physrevx 2024

  14. [14]

    Journal of Fluid Mechanics433, 383–409 (2001) https://doi.org/ 10.1017/S0022112001003597

    Renner, C., Peinke, J., Friedrich, R.: Experimental indications for markov properties of small-scale turbulence. Journal of Fluid Mechanics433, 383–409 (2001) https://doi.org/ 10.1017/S0022112001003597

    work page doi:10.1017/s0022112001003597 2001

  15. [15]

    Stats6(2), 626–642 (2023) https://doi.org/10.3390/stats6020040

    Song, J., Ma, M.: Climate change: Linear and nonlinear causality analysis. Stats6(2), 626–642 (2023) https://doi.org/10.3390/stats6020040

    work page doi:10.3390/stats6020040 2023

  16. [16]

    Petelczyc, M., ˙Zebrowski, J.J., Orłowska-Baranowska, E.: A fixed mass method for the kramers-moyal expansion—application to time series with out- liers. Chaos: An Interdisciplinary Journal of Nonlinear Science25(3), 033115 (2015) https://doi.org/10.1063/1.4914547 https://pubs.aip.org/aip/cha/article- pdf/doi/10.1063/1.4914547/14609149/033115 1 online.pdf

    work page doi:10.1063/1.4914547 2015

  17. [17]

    Journal of the American Statistical Association74(366), 427–431 (1979)

    Dickey, D.A., Fuller, W.A.: Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association74(366), 427–431 (1979)

    work page 1979

  18. [18]

    Journal of Econometrics54(1–3), 159–178 (1992)

    Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., Shin, Y .: Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics54(1–3), 159–178 (1992)

    work page 1992

  19. [19]

    Understanding Complex Systems, p

    Rahimi Tabar, M.R.: Analysis and Data-Based Reconstruction of Complex Nonlinear 11 Dynamical Systems, 1st edn. Understanding Complex Systems, p. 280. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-18472-8

    work page doi:10.1007/978-3-030-18472-8 2019

  20. [20]

    Scientific Reports6(1), 35435 (2016) https://doi.org/10

    Anvari, M., Tabar, M.R.R., Peinke, J., Lehnertz, K.: Disentangling the stochastic behav- ior of complex time series. Scientific Reports6(1), 35435 (2016) https://doi.org/10. 1038/srep35435

    work page 2016

  21. [21]

    Springer Series in Synergetics, p

    Gardiner, C.: Stochastic Methods, 4th edn. Springer Series in Synergetics, p. 447. Springer, Berlin, Heidelberg (2009)

    work page 2009

  22. [22]

    Wilson, G.T.: Time series analysis: Forecasting and control, 5th edition, by george e. p. box, gwilym m. jenkins, gregory c. reinsel, and greta m. ljung, 2015. published by john wiley and sons inc., hoboken, new jersey, pp. 712. isbn: 978-1-118-67502-

    work page 2015

  23. [23]

    Journal of Time Series Analysis37(5), 709–711 (2016) https://doi.org/10.1111/jtsa. 12194

    work page doi:10.1111/jtsa 2016

  24. [24]

    Journal of Time Series Analysis13(6), 517–525 (1992)

    Gong, G., Tong, H.: Testing the markov property of a time series using conditional entropies. Journal of Time Series Analysis13(6), 517–525 (1992)

    work page 1992

  25. [25]

    Entropy23, 517 (2021) https://doi.org/10

    Rydin Gorj ˜ao, L., Witthaut, D., Lehnertz, K., Lind, P.G.: Arbitrary-order finite-time corrections for the kramers–moyal operator. Entropy23, 517 (2021) https://doi.org/10. 3390/e23050517

    work page 2021

  26. [26]

    Springer Series in Synergetics

    Risken, H.: The Fokker-Planck Equation: Methods of Solution and Applications, 2nd edn. Springer Series in Synergetics. Springer, Berlin (1996). https://doi.org/10.1007/ 978-3-642-61544-3 12 Appendix A Stationarity test Stationarity is essential for time-homogeneous KM analysis, ensuring the validity of time- invariant stochastic coefficientsD (n)(x)[16]. ...

    work page 1996