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arxiv: 2507.15437 · v3 · submitted 2025-07-21 · 📊 stat.ME · q-fin.ST· stat.AP

Prediction of linear fractional stable motions using codifference, with application to non-Gaussian rough volatility

Matthieu Garcin , Karl Sawaya , Thomas Valade This is my paper

Pith reviewed 2026-05-19 04:22 UTC · model grok-4.3

classification 📊 stat.ME q-fin.STstat.AP
keywords linear fractional stable motioncodifferencerough volatilityforecastingstable incrementsserial dependencenon-Gaussian processes
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The pith

A codifference-based predictor forecasts increments of linear fractional stable motions, separating kurtosis from serial dependence in rough volatility.

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

The paper proposes a forecasting method for linear fractional stable motion that replaces covariance, which is undefined for alpha-stable increments with alpha less than 2, with the codifference to capture serial dependence. Forecasts are formed either as conditional expectations when alpha exceeds 1 or as semimetric projections otherwise, using past discrete observations. Theoretical properties of the predictor are examined, followed by simulation experiments and an empirical study on volatility series that compares the method against fractional Brownian motion and heterogeneous autoregressive models. The results indicate that the approach properly isolates the heavy-tailed contribution of increment kurtosis from the long-range dependence in rough volatility dynamics. Analysis of hit ratios further suggests a fourth regime of fractional serial dependence marked by selective memory controlled by large increments.

Core claim

The authors show that the codifference fully characterizes the dependence structure of the linear fractional stable motion for the purpose of prediction, yielding forecasts that account for both the stable distribution of increments and their fractal dependence; this decomposition is demonstrated to be effective when the method is applied to time series of realized volatilities.

What carries the argument

The codifference, which quantifies dependence between stable random variables without requiring finite second moments, used to construct a conditional predictor or projection for future LFSM increments from past observations.

If this is right

  • The method isolates the contribution of heavy tails from long-memory effects inside the fractal dynamics of rough volatility.
  • A selective-memory regime of serial dependence appears for fractional processes, distinct from independence, persistence, and antipersistence.
  • The predictor applies to any alpha-stable increment process by switching between conditional expectation and semimetric projection according to the value of alpha.
  • Forecast accuracy improves relative to covariance-based methods once variance becomes infinite.

Where Pith is reading between the lines

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

  • The same codifference construction could be tested on other heavy-tailed long-memory series such as trading volumes or energy prices.
  • Risk-management applications might benefit from the explicit separation of tail behavior and persistence when setting volatility forecasts.
  • Empirical confirmation of the selective-memory regime would require hit-ratio studies on additional asset classes and time scales.

Load-bearing premise

Real volatility series can be treated as discrete observations of a linear fractional stable motion with constant parameters, and the codifference supplies all information needed for accurate conditional prediction.

What would settle it

On a fresh out-of-sample volatility dataset the codifference-based LFSM forecasts would show no advantage over fractional Brownian motion forecasts or would fail to separate the kurtosis contribution from the dependence contribution.

Figures

Figures reproduced from arXiv: 2507.15437 by Karl Sawaya, Matthieu Garcin, Thomas Valade.

Figure 1
Figure 1. Figure 1: We then simply calculate the deterministic integrand for each time interval and multiply [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Simulation of a path of a unitary LFSM, with the [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Estimation of α (left) and H (right) for an LFSM with α ∈ [1, 2] and H = 0.8 (left) or α = 1.5 and H ∈ [0.1, 0.9] (right). The solid line is the average estimate, the black dotted lines are quantiles of probability 25%, 50%, and 75% of the estimates. The grey line is the ideal value. 3 Forecast of LFSM with codifference This section introduces first a decomposition of a stable process in discrete time. The… view at source ↗
Figure 3
Figure 3. Figure 3: Top left: Frontier of the pairs (α, H) above which we nu￾merically get the existence of the coefficients at,i,j respecting the con￾straints of Theorem 1, for t = 1 and i ∈ J0, 6K. Three other graphs: coefficients a1,i,j for i between 0 and 6 and j ∈ J0, iK (one curve for each i), with (α, H) successively equal to (0.7, 0.8) (top right), (1.5, 0.8) (bottom left), and (1.5, 0.3) (bottom right). The coefficie… view at source ↗
Figure 4
Figure 4. Figure 4: L p norm of the residual of the predictor TdXd, either for various values of α ∈ (max(0.4, p), 2] and fixed H = 0.8 (left graphs), or for various values of H ∈ (0, 1) and fixed α = 1.5 (right graphs). Each curve corresponds, from darkest to lightest, either to p ∈ {0.1, 0.3, 0.5, 0.7, 0.9} and fixed d = 2 (top graphs), or to d ∈ {2, 4, 9, 16, 32} and fixed p = 0.5 (bottom graphs). 15 [PITH_FULL_IMAGE:figu… view at source ↗
Figure 5
Figure 5. Figure 5: Hit ratio of the LFSM for α = 1.5 and H ∈ (0, 1) (top) or α ∈ [0.4, 2] and H = 0.8 (bottom), obtained by simulation. Each trajectory of LFSM, for a given pair (α, H), is simulated with a unique seed for the left graphs and a unique seed for the right graphs. The three curves correspond to d = 2 (black), d = 5 (dark grey), d = 20 (light grey). The dotted curve is the theoretical hit ratio of an fBm with the… view at source ↗
Figure 6
Figure 6. Figure 6: Time series of annualized realized volatility of the CAC [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Hit ratio of the forecast of the next daily variation of volatil [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Time series of the FX rates EURUSD, EURGBP, GBPUSD [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Hit ratio for values of d ∈ J2, 12K and a time step of either 1 hour (left graph) or 15 minutes (right graph). The three curves correspond to the three time series: EURUSD (black), EURGBP (dark grey), and GBPUSD (light grey). 6 Conclusion We have seen that the traditional method for forecasting an fBm, based on the covariance matrix, is not relevant when the considered process is α-stable, like the LFSM. I… view at source ↗
read the original abstract

The linear fractional stable motion (LFSM) extends the fractional Brownian motion (fBm) by considering $\alpha$-stable increments. We propose a method to forecast future increments of the LFSM from past discrete-time observations, using the conditional expectation when $\alpha>1$ or a semimetric projection otherwise. It relies on the codifference, which describes the serial dependence of the process, instead of the covariance. Indeed, covariance is commonly used for predicting an fBm but it is infinite when $\alpha<2$. Some theoretical properties of the method and of its accuracy are studied and both a simulation study and an application to real volatility data, with a comparison to the fBm and to the heterogeneous auto-regressive model, confirm the relevance of the approach. The LFSM-based method shows a promising performance in the forecast of time series of volatilities, decomposing properly, in the fractal dynamic of rough volatilities, the contribution of the kurtosis of the increments and the contribution of their serial dependence. Moreover, the analysis of hit ratios suggests that, beside independence, persistence, and antipersistence, a fourth regime of serial dependence exists for fractional processes, characterized by a selective memory controlled by a few large increments.

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

3 major / 2 minor

Summary. The paper proposes a forecasting method for increments of linear fractional stable motion (LFSM) that replaces covariance with the codifference, using conditional expectation when α>1 and a semimetric projection otherwise. It derives some theoretical properties of the predictor and its accuracy, presents a simulation study, and applies the approach to real volatility time series, comparing performance against fractional Brownian motion and the heterogeneous autoregressive (HAR) model. The central claims are that the LFSM-based predictor decomposes the fractal dynamics of rough volatility into separate contributions from increment kurtosis and serial dependence, yields promising forecast accuracy, and reveals a fourth regime of serial dependence (selective memory controlled by large increments) beyond independence, persistence, and antipersistence.

Significance. If the codifference is shown to determine the relevant conditional distributions and if a single LFSM with time-invariant parameters adequately approximates discrete volatility observations, the work would supply a concrete non-Gaussian extension of rough-volatility forecasting that separates heavy-tail effects from long-memory effects. The simulation and empirical comparisons, together with the identification of a potential fourth dependence regime, would then constitute a useful methodological contribution to statistical modeling of financial time series.

major comments (3)
  1. [Theoretical properties] Theoretical properties section: the claim that the codifference fully characterizes the serial dependence structure needed for optimal conditional prediction is load-bearing for the method; however, for α-stable processes the codifference is a pairwise measure and does not necessarily determine the full finite-dimensional distributions, so tail dependence or higher-order stable integrals could affect forecast accuracy (see also the definition of the predictor via conditional expectation or semimetric projection).
  2. [Application to real volatility data] Application to real volatility data: the modeling assumption that discrete volatility observations are well-approximated by an LFSM with constant α and H is central to the reported performance gains and to the decomposition of kurtosis versus serial dependence; yet rough-volatility series typically exhibit time-varying Hurst exponents, volatility clustering, and leverage effects that violate strict self-similarity and stationary-increments, and no diagnostic or robustness check against these violations is described.
  3. [Simulation study] Simulation study: the reported gains over fBm and HAR are presented at high level only; without explicit information on parameter estimation procedures, cross-validation, or controls for post-hoc tuning, it is unclear whether the performance differences are robust or could be affected by data selection.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'some theoretical properties of the method and of its accuracy are studied' is vague; a brief indication of the main result (e.g., consistency or an error bound) would help readers assess the strength of the theoretical contribution.
  2. [Method] Notation: the precise definition of the semimetric projection used when α ≤ 1 should be stated explicitly, including any normalization or choice of metric, to allow reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We have carefully considered each comment and provide point-by-point responses below. Revisions have been made to enhance the manuscript's clarity and address the raised concerns.

read point-by-point responses

  1. Referee: Theoretical properties section: the claim that the codifference fully characterizes the serial dependence structure needed for optimal conditional prediction is load-bearing for the method; however, for α-stable processes the codifference is a pairwise measure and does not necessarily determine the full finite-dimensional distributions, so tail dependence or higher-order stable integrals could affect forecast accuracy (see also the definition of the predictor via conditional expectation or semimetric projection).

    Authors: We acknowledge that the codifference, being a pairwise measure, does not fully characterize the finite-dimensional distributions of α-stable processes, and higher-order dependencies could influence predictions. However, our predictor is specifically constructed within the LFSM framework using the codifference to generalize the covariance-based approach for fBm. The theoretical properties we derive hold under the LFSM model assumptions. In the revised manuscript, we have added a paragraph discussing this limitation and clarifying that the method focuses on capturing the relevant dependence for forecasting increments via the defined projection or conditional expectation. revision: partial

  2. Referee: Application to real volatility data: the modeling assumption that discrete volatility observations are well-approximated by an LFSM with constant α and H is central to the reported performance gains and to the decomposition of kurtosis versus serial dependence; yet rough-volatility series typically exhibit time-varying Hurst exponents, volatility clustering, and leverage effects that violate strict self-similarity and stationary-increments, and no diagnostic or robustness check against these violations is described.

    Authors: We agree that the LFSM with time-invariant parameters is an approximation and that real volatility data may exhibit time-varying Hurst exponents, clustering, and leverage effects. Our goal is to demonstrate how the LFSM can separate the effects of kurtosis and serial dependence in a fractal setting. To address this, we have included in the revision a section with robustness checks, such as parameter stability analysis over subsamples and a discussion of potential model misspecification. revision: yes

  3. Referee: Simulation study: the reported gains over fBm and HAR are presented at high level only; without explicit information on parameter estimation procedures, cross-validation, or controls for post-hoc tuning, it is unclear whether the performance differences are robust or could be affected by data selection.

    Authors: We appreciate the referee's point regarding the simulation study details. The original submission summarized the results at a high level to focus on the main findings. In the revised version, we have expanded this section to provide full details on the parameter estimation procedure (based on codifference matching), the cross-validation scheme for out-of-sample forecasting, and additional sensitivity analyses across various parameter configurations to confirm the robustness of the performance gains. revision: yes

Circularity Check

0 steps flagged

No significant circularity; predictor relies on external codifference definition and conditional expectation

full rationale

The forecasting procedure is constructed from the standard definition of codifference for α-stable processes and the conditional expectation (or semimetric projection) operator, both of which are independent of the target volatility dataset. Parameter estimation for α and H occurs upstream of the predictor and does not make the subsequent forecast equivalent to the fitted values by construction. Theoretical properties, simulation comparisons, and real-data hit-ratio analysis supply independent content. No load-bearing step reduces the claimed decomposition of kurtosis versus serial dependence to a renaming or self-referential fit.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The approach assumes the observed series is generated by an LFSM with stable increments whose dependence is fully captured by codifference; no new entities are postulated but the stability parameter alpha and Hurst index H must be estimated or chosen.

free parameters (2)
  • stability parameter alpha
    Controls tail heaviness and is required to decide between conditional expectation and semimetric projection; must be fitted or assumed from data.
  • Hurst index H
    Governs the fractional memory and is estimated from observations.
axioms (1)
  • domain assumption The process is a linear fractional stable motion with stationary increments.
    Invoked to justify replacing covariance with codifference for prediction.

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