Stochastic Volterra equations: failure of the time-homogeneous Markov property

Kristof Wiedermann; Martin Friesen; Stefan Gerhold

Stochastic Volterra equations: failure of the time-homogeneous Markov property

T0 review · 0 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Stochastic Volterra equations are Markov only for exponential kernels

desk verdict The affine characterization is appealing and likely correct, but Theorem 2.3's resolvent sign error makes the zero-drift linear case unproven as written. read the letter →

arxiv 2510.22416 v2 pith:42JTT5S2 submitted 2025-10-25 math.PR

Martin Friesen , Stefan Gerhold , Kristof Wiedermann This is my paper · ORCID

classification math.PR MSC 60G1560G2260H2060J25

keywords stochasticVolterraequationtime-homogeneousMarkovpropertykernelroughvolatilityaffineprocessfractionalRiemann-LiouvilleGaussiansmall-timecentrallimittheorem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to prove what practitioners have long assumed: stochastic Volterra equations, whose dynamics are shaped by a memory kernel, are not time-homogeneous Markov processes except in a narrow degenerate case. For equations with affine drift, it proves a sharp classification: if the solution family is time-homogeneous Markov, the Volterra kernel must be K(t)=c e^{-λt}, an exponential tied to the initial curve; for constant initial curves this means only the constant kernel, i.e. classical SDEs, qualifies. For Hölder-continuous coefficients, it proves a transfer theorem: under small-time power-law and CLT assumptions, non-Markovianity of the Gaussian process K*dB — which holds for fractional Riemann-Liouville kernels with H≠1/2 — implies non-Markovianity of the SVE. A sympathetic reader should care because rough volatility models and other path-dependent models are built on these equations, and their non-Markovianity is what makes them hard to simulate and hedge.

What carries the argument

Two mechanisms carry the argument. First, the resolvent kernel E_K, the solution of E_K=K+β K*E_K, converts the affine SVE's mean into an explicit Volterra formula; Markovianity then forces the composed term E^λ_1 to be exponential, and the resolvent equation propagates this to K(t)=K(0)e^{-λt}. Second, the small-time central limit theorem rescales X_{t/n} by the inverse L^2 norm of K on [0,1/n]; if the SVE were Markov, the scaled process would converge to a Markov Gaussian process, so it suffices to test the Gaussian convolution K*dB. The test is the classical covariance condition for centered Gaussian processes, c(s,u)c(t,t)=c(s,t)c(t,u), and for the fractional Riemann-Liouville kernel the

What would settle it

Exhibit a single non-exponential continuous kernel K and coefficients satisfying the paper's assumptions for which the SVE (1.2) genuinely satisfies the time-homogeneous Markov property (1.3); Theorem 2.1 says no such example exists, so a valid example would refute the classification. Alternatively, find a kernel satisfying Assumption 4.1 for which K*dB is non-Markov but the SVE is Markov — Theorem 4.2 says this too is impossible.

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Extended reading notes

Core claim

The central claim is Theorem 2.1: if a stochastic Volterra equation with affine drift b(x)=b_0+βx admits a continuous weak solution for every x in a closed domain D, and those laws satisfy the time-homogeneous Markov property (1.3), then the Volterra kernel must be K(t)=c e^{-λt} for all t, where λ is the exponential rate of the initial curve xe^{-λt}. Markovianity forces the first moment to satisfy a flow identity; comparing the direct moment formula with the Markov composition leads to the multiplicative relation E^λ_1(T)=E^λ_1(T-t)E^λ_1(t), whose only continuous solution is exponential, and the resolvent equation then propagates that exponential form to K. Theorem 4.2 extends this to Höld

Load-bearing premise

The broad non-Markov theorem rests on an unproved small-time central limit theorem from a companion preprint; if that CLT or its exponent conditions fail, the reduction of SVEs to Gaussian processes collapses.

Editorial extensions

If this is right

For affine Volterra processes — Volterra Ornstein-Uhlenbeck, square-root (rough CIR), and Jacobi processes — the Markov property forces K(t)=c e^{-λt}; the rough CIR equation with a fractional kernel is therefore non-Markov.
With a constant initial curve (λ=0), the only Markovian SVEs among affine-drift equations are classical SDEs with constant kernel K≡c.
For Hölder coefficients, any kernel satisfying Assumption 4.1 whose scaled limit K*dB is non-Markov (e.g. fractional kernels H≠1/2, log-modulated fractional kernels) yields non-Markov SVEs, irrespective of the drift.
Exponential kernels are genuinely Markovian: they reduce the SVE to the semimartingale SDE dX_t=(K(0)b(X_t)-λX_t)dt+K(0)σ(X_t)dB_t.
The small-time CLT route is restricted to time-homogeneous Markovianity; proving failure of the time-inhomogeneous property would require a conditional CLT at arbitrary times, as the authors note.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A practical test emerges: before modelling with a candidate kernel, one can check the covariance condition for the Gaussian convolution K*dB; if that condition fails, the SVE cannot be Markov under the CLT assumptions.
The first-moment flow identity used in Theorem 2.1 gives a necessary consistency condition for any candidate transition kernel of an affine Volterra process, going beyond Markovianity itself.
The result suggests that Markovian approximation schemes for rough volatility are approximations in a strict sense: the true law is non-Markov, so any finite-dimensional Markov lift can only approximate, not reproduce, the path measure.
Extending the small-time CLT to conditional laws at positive times would turn Theorem 4.2 into a statement about time-inhomogeneous Markovianity, which is what many numerical methods actually assume.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read the paper on stochastic Volterra equations and the Markov property. The headline: the affine characterization is likely correct, but Theorem 2.3 has a genuine proof error that needs fixing, and the paper leans on an unpublished CLT from the same group.

What's actually new and good: Theorem 2.1 is the core result. It shows that for affine drift with b0 or β nonzero, the time-homogeneous Markov property forces the kernel to be exponential, using first-moment flow equations. The proof is transparent and self-contained. Theorem 2.2 handles the zero-drift square-root case and uses second moments instead. Section 3 gives a novel direct proof that the fractional Riemann-Liouville process is not Markov via conditional Gaussian means, though the fact itself was already in Bauer–Gerhold. Theorem 4.2 is a plausible reduction: if the small-time CLT from [18] holds, non-Markovianity of the Gaussian convolution implies non-Markovianity of the Hölder-coefficient SVE. That's a useful tool, but it depends on the authors' own preprint and Assumption 4.1 is stated, not proved, here.

The soft spot is real: in Theorem 2.3, equation (2.18) has the wrong sign. The second moment solves m = f − K*m with K = −σ0^2 K^2. The resolvent of K gives m = f + R_K*f, not f − R_K*f. A quick test with λ=0, σ0=1, K≡1, x=1 yields e^T as the true moment, while (2.18) gives 2−e^T. So the proof of Theorem 2.3 collapses as written. The conclusion may still be true — flipping the sign seems to convert the subsequent equations into the correct multiplicative form — but this is not a typo. It is a load-bearing algebraic error, and the authors need to redo that part.

Secondary concerns: Section 3's non-Markovianity is not new, and Theorem 4.2 rests on the self-cited [18]. These are worth noting but not fatal.

Who this is for: anyone working on Markovianity of Volterra processes and rough volatility. The moment approach is elegant and the question is folklore. The paper deserves refereeing, but the current version needs major revision, mainly fixing Theorem 2.3.

Referee Report

0 major / 5 minor

Summary. The paper studies one-dimensional stochastic Volterra equations (SVEs) of the form X_t = x e^{-λt} + ∫_0^t K(t-s)b(X_s)ds + ∫_0^t K(t-s)σ(X_s)dB_s and asks when the associated family of laws can satisfy the time-homogeneous Markov property (1.3). The main positive results are necessary conditions on the Volterra kernel. Theorem 2.1 shows that for affine drift b(x)=b0+βx, under weak existence and a technical domain condition, Markovianity forces K(t)=ce^{-λt}. Theorems 2.2 and 2.3 treat the zero-drift cases σ(x)=σ0√x and σ(x)=σ0x, again concluding K(t)=ce^{-λt}. Section 3 gives a direct Gaussian computation proving that the fractional Riemann-Liouville process ∫_0^t (t-s)^{H-1/2}dB_s is not Markov for H≠1/2. Section 4, under Assumption 4.1 and using a small-time CLT from the authors' preprint [18], proves Theorem 4.2: if the limiting Gaussian process K*dB is not a time-homogeneous Markov process, then the SVE with Hölder coefficients cannot have the time-homogeneous Markov property. The paper is clearly written and the affine part is self-contained.

Significance. If the results hold, the paper provides the first rigorous and fairly general non-Markovity criterium for stochastic Volterra equations, a fact that is often taken for granted but rarely proved. The strength of the paper lies in the explicit moment computations: Theorems 2.1–2.3 reduce Markovianity to functional equations for resolvents and moments, and the conclusions are sharp for the examples (Volterra OU, Volterra square-root, Jacobi Volterra). Section 3's direct conditional-Gaussian proof is elegant and self-contained, and it gives an explicit quantitative signal (the asymptotic conditional mean of order τ^{1-2H}) rather than relying only on the Doob criterion. The assumptions of Section 4 are stated transparently, and the authors acknowledge the dependence on the companion preprint [18]. The main caveat is that Theorem 4.2 is only as reliable as that external CLT; this does not affect Theorems 2.1–2.3 or Section 3.

minor comments (5)

[Section 4 / Theorem 4.2] The proof of Theorem 4.2 invokes [18, Theorem 2.2] as a black box, and Assumption 4.1 is essentially a restatement of its hypotheses. Since this is one of the two advertised approaches and the only one covering general Hölder coefficients, the manuscript would be easier to verify if the exact statement of the needed CLT were reproduced, or if Theorem 4.2 were explicitly labelled as conditional on [18]. This is a self-containedness/verifiability issue, not an internal inconsistency.
[Theorem 2.3, Eq. (2.18)] I checked the potential sign issue in (2.18). With the resolvent convention used in [21] for the equation y=f−k*y, the resolvent r satisfies r=k−k*r and the solution is y=f−r*f. For k=−σ0^2K^2, formula (2.18) is correct, and the subsequent equations (2.19)–(2.22) are consistent. To avoid confusion, however, the authors should state this resolvent convention explicitly, since the notation R_K for the resolvent of −σ0^2K^2 collides with the earlier R_K for the affine drift problem.
[Abstract and Section 2] The abstract says that for affine drifts the Markov property only holds for exponential kernels. In the zero-drift case b≡0, which is affine, Theorems 2.2 and 2.3 cover only σ(x)=σ0√x and σ(x)=σ0x. The wording is slightly overbroad; suggest adding 'in the cases considered' or listing the covered diffusions.
[Lemma 3.1] The parametrization (β3,β2,β1)(τ)=(τ,τ^2,τ^3) is confusing because the displayed order is not increasing. Writing β1=τ^3, β2=τ^2, β3=τ with 0<τ<1 would make the ordering 0<β1<β2<β3 easier to read.
[Section 4, Example 4.4] The statement that the log-modulated fractional kernel satisfies (4.1) with γ*=H and arbitrary γ∈(0,H) is correct for small t, but the verification is only sketched via Karamata's theorem and a pointer to [18, Example 2.8]. Adding the short computation would make the example self-contained.

Circularity Check

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No significant circularity: Section 2's moment comparison is self-contained, and Section 4's dependence on the authors' CLT preprint is an external conditional input, not an assumed target result.

full rationale

The central derivations are not circular. In Section 2, the authors assume the time-homogeneous Markov property (1.3) and derive explicit first- and second-moment identities from the SVE using standard Volterra variation-of-constants and resolvent theory. Comparing the moment formula (2.3) with the Markov-implied iterated expectation (2.4) yields functional equations such as (2.5), (2.6), and (2.9); solving these equations forces the kernel to be exponential. The auxiliary functions E_K and R_K are defined from the given kernel K, not from the conclusion, so no 'prediction' is equivalent by construction to a fitted input. The zero-drift cases Theorems 2.2 and 2.3 set up deterministic linear Volterra equations for second moments and solve them via the resolvent, again without importing the target exponential form. Section 3 is an independent direct Gaussian calculation: Lemma 3.1 computes the conditional mean of the Riemann-Liouville process and shows it is asymptotically nonzero for H != 1/2, proving non-Markovianity without relying on the earlier self-cited fact in [8]. Section 4 is a conditional reduction theorem: its external input is the small-time CLT from the authors' preprint [18], whose assumptions in Assumption 4.1 do not include the Markov property or the target conclusion. The proof shows that if the SVE were Markov, the CLT limit Y^{sigma,0} would inherit Markov transition kernels, contradicting the assumed non-Markovianity of the Gaussian process. This is a standard contrapositive reduction, not a circular step. The paper explicitly discloses the reliance on [18] and the restriction to t=0. The alleged sign error in Theorem 2.3 is a mathematical correctness concern, not a circularity concern. No step in the derivation chain reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles, forces, or fitted constants. Its central claims depend on standard Volterra-resolvent and Gaussian-process theory, on external weak-existence results, and on the authors' own small-time CLT [18], which is stated as an assumption rather than proved in the text. No data are used and no parameters are fitted to observations.

assumptions (6)

domain assumption Weak existence of continuous D-valued solutions for every x∈D, as cited from [1,4,13,26] and [4, Thm 3.4/3.6].
Theorems 2.1–2.3 and 4.2 are statements about the family of laws (P^x); the paper assumes existence of these laws rather than proving it in full.
domain assumption Small-time CLT of [18, Theorem 2.2] for SVEs.
Provides the scaling limit h_n(X_{·/n}) ⇒ Y^{σ,0} used in Step 1 of Theorem 4.2; the proof is not reproduced in this paper and relies on a preprint by the same authors.
standard math Doob's covariance criterion and its Mehr–McFadden formulation for Gaussian Markov processes.
Used in Section 3 and in the interpretation of the Gaussian limit process in Theorem 4.2.
standard math Volterra resolvent theory, especially [21, Theorems 2.3.1 and 2.3.5].
Used in Section 2 to define E_K and R_K and to solve the linear Volterra equations for moments.
standard math Hypergeometric integral representations, Pfaff's transformation, and Gauss's summation formula from [7].
Used in Lemma 3.1 to compute the small-τ asymptotics of the covariance entries θ_ij.
standard math Karamata's theorem / regular variation theory from [12].
Used in Example 4.4 to verify Assumption 4.1 for the log-modulated fractional kernel.

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Cite this review

Pith. "Pith review of Stochastic Volterra equations: failure of the time-homogeneous Markov property." pith.science (2026). https://pith.science/paper/42JTT5S2

@misc{pith2026251022416,
  author       = {Pith},
  title        = {Pith review of: Stochastic Volterra equations: failure of the time-homogeneous Markov property},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/42JTT5S2}},
  note         = {Machine review of arXiv:2510.22416}
}

read the original abstract

Path-dependence is a defining feature of many real-world systems, with applications ranging from population dynamics to rough volatility models and electricity spot prices. In stochastic Volterra equations (SVEs), such dependence is encoded in the Volterra kernel, which dictates how past trajectories influence present dynamics on infinitesimal time scales. This structure suggests a breakdown of the Markov property. In this article, we develop computational techniques and methods based on small-time asymptotics for SVEs with H\"older coefficients to rigorously establish that they cannot possess the time-homogeneous Markov property. In particular, for affine drifts, we characterise the time-homogeneous Markov property and show that under natural non-degeneracy conditions, it only holds for exponential Volterra kernels.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Abi Jaber, C

[1]E. Abi Jaber, C. Cuchiero, M. Larsson, and S. Pulido,A weak solution theory for stochastic Volterra equations of convolution type, The Annals of Applied Probability, 31 (2021), pp. 2924 –

2021

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Gatheral, T

[19]J. Gatheral, T. Jaisson, and M. Rosenbaum,Volatility is rough, Quantitative Finance, 18 (2018), pp. 933–949. [20]S. Gerhold,Asymptotic analysis of a double integral occurring in the rough Bergomi model, Mathe- matical Communications, 25 (2020), pp. 171–184. [21]G. Gripenberg, S.-O. Londen, and O. Staffans,Volterra integral and functional equations, vo...

2018

[3] [37]

Abi Jaber and O

[3]E. Abi Jaber and O. El Euch,Multifactor Approximation of Rough Volatility Models, SIAM Journal on Financial Mathematics, 10 (2019), pp. 309–349. [4]E. Abi Jaber, M. Larsson, and S. Pulido,Affine Volterra processes, The Annals of Applied Probability, 29 (2019), pp. 3155 –

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[4] [1953]

El Euch and M

[16]O. El Euch and M. Rosenbaum,Perfect hedging in rough Heston models, The Annals of Applied Probability, 28 (2018), pp. 3813–3856. [17]O. El Euch and M. Rosenbaum,The characteristic function of rough Heston models, Mathematical Finance, 29 (2019), pp. 3–38. [18]M. Friesen, S. Gerhold, and K. Wiedermann,Small-time central limit theorems for stochastic Vo...

arXiv 2018

[5] [1987]

Ca ˜nadas and M

[13]O. Ca ˜nadas and M. Friesen,Comparison principles for stochastic Volterra equations. Preprint, available athttps://arxiv.org/abs/2403.15801,

arXiv

[6] [1998]

[23]C. B. Mehr and J. A. McF adden,Certain properties of Gaussian processes and their first-passage times, The Journal of the Royal Statistical Society, Series B, 27 (1965), pp. 505–522. [24]I. Nourdin,Selected Aspects of Fractional Brownian Motion, Bocconi & Springer Series, Springer Milano,

1965

[7] [1999]

Bauer and S

[8]B. Bauer and S. Gerhold,Self-similar Gaussian Markov processes, To appear in Stochastics, (2025), pp. 1–19. Available athttps://doi.org/10.1080/17442508.2025.2540533. [9]C. Bayer and S. Breneis,Markovian approximations of stochastic Volterra equations with the fractional kernel, Quantitative Finance, 23 (2023), pp. 53–70. [10]C. Bayer, P. K. Friz, M. F...

arXiv 2025

[8] [2013]

[25]D. J. Pr ¨omel and D. Scheffels,Stochastic Volterra equations with H¨ older diffusion coefficients, Stochastic Processes and their Applications, 161 (2023), pp. 291–315. [26]D. J. Pr¨omel and D. Scheffels,On the existence of weak solutions to stochastic Volterra equations, Electronic Communications in Probability, 28 (2023), pp. 1 –

2023