Convergence in H\"older norms for Markovian approximations of stochastic Volterra equations

Noé Corneille; Kristin Kirchner; Pietro Pezzoli Frigerio

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Convergence in H\"older norms for Markovian approximations of stochastic Volterra equations

Noé Corneille, Kristin Kirchner, Pietro Pezzoli Frigerio

arXiv:2609.17472v1Sep 15, 2026math.PRcs.NAmath.NA

TL;DR

Stochastic Volterra processes retain memory through their entire history, and prior work had not analyzed Markovian approximation errors in pathwise Hölder norms. This paper bounds errors between processes with different kernels and constructs sinc-based Markovian approximations for fractional kernels, with convergence exponential in the square root of the node count in the stated norms.

Problem
Volterra increments depend on the process’s entire history, and pathwise Hölder errors for Markovian approximations had not been analyzed, to the authors’ knowledge.
Source: §1.1 · p. 1, §1.2 · p. 2
Method
The paper bounds differences between processes with the same Lipschitz coefficients and different kernels, then uses sinc quadrature to construct Markovian approximations for fractional kernels.
Source: §1.3 · p. 3
Results
The fractional-kernel Markovian approximations converge exponentially in the square root of the quadrature-node count in pointwise, pathwise Lebesgue, and Hölder norms.
Source: §1.3 · p. 3, §5.2 · p. 27
Takeaways & Limitations
The Hölder convergence range reaches essentially the maximal regularity permitted by the exact stochastic Volterra process.
Source: §1.3 · p. 3
The Hölder integrability criterion requires β⋆>0, so the pointwise and Hölder bounds are treated separately.
Source: §5.3 · p. 29

Abstract

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We bound the difference between two stochastic Volterra processes with identical Lipschitz coefficients but different kernels. For non-convolution kernels, we establish estimates in $C^0([0,T];L^p(\Omega))$, $p\geq 2$, and for convolution kernels in $L^p(\Omega;L^q(0,T))$, $q \in [1,p]$, and $C^\beta([0,T];L^p(\Omega))$, $L^p(\Omega;C^\beta([0,T]))$, where the range of the H\"older exponent $\beta \in (0,1]$ is the maximal permitted by the regularity of the processes. For the fractional kernel, we then construct Markovian approximations whose error we show to decay as $e^{-a\sqrt{N}}$ in the aforementioned norms, using an $N$-node quadrature based on sinc methods. Numerical experiments for the fractional Brownian motion verify our findings.

1. Introduction

Stochastic Volterra processes model memory-dependent phenomena, but their non-locality makes classical time-stepping computationally expensive. This paper develops error bounds for processes with different kernels and sinc-based Markovian approximations, including convergence in Hölder norms.

1.1. Background and motivation.

The entire history of a Volterra process affects its increments, making classical approximation approaches computationally expensive.The kernels give the process its non-local structure, unlike an SDE whose future evolution depends on its current state.Source: §1.1 · p. 1

1.2. Previous work.

Earlier work analyzed Markovian approximation errors, but the pathwise error in Hölder norms had not been investigated, to the authors’ knowledge.Existing time-stepping schemes also have quadratic cost in the number of time-grid points, while Gaussian quadrature methods may require recomputing nodes and weights as N increases.Source: §1.2 · pp. 2–3

1.3. Contributions.

The paper bounds differences between Volterra processes with identical Lipschitz coefficients and different kernels, including non-convolution kernels and pathwise Hölder norms.For fractional kernels, it constructs sinc-quadrature Markovian approximations with error exponential in the square root of the number of nodes, up to the maximal Hölder regularity permitted by the exact process.Source: §1.3 · p. 3

1.4. Outline.

The paper proceeds from notation and assumptions to error estimates, fractional-kernel quadrature, Markovian convergence results, and numerical experiments.The experiments aim to verify the quadrature and strong-convergence results for fractional kernels.Source: §1.4 · p. 3

2. Preliminaries

The preliminaries establish notation and the setting for stochastic Volterra processes. They specify coefficient and kernel assumptions used in the later analysis.

2.1. Notation.

The notation section defines function spaces including C^0 and C^β spaces, and fixes conventions for integers, indicators, and the probability space.The C^β norm combines the supremum norm with the β-Hölder seminorm.Source: §2.1 · p. 4

2.2. Definitions and setting.

A continuous stochastic Volterra process is adapted, has continuous sample paths, and satisfies the Volterra equation with specified initial value, coefficients, and kernels.For convolution kernels, the kernels depend on t and s through t−s when t>s and vanish when t≤s.Source: §2.2 · p. 4
The setting assumes Lipschitz coefficients and imposes integrability and increment regularity conditions on the kernels.For convolution kernels, nonnegative non-increasing kernels are additionally controlled by L^1 and L^2 bounds that scale as t^α.Source: §2.2 · p. 5

3. Difference in H¨older norms of stochastic Volterra processes

This section establishes existence, uniqueness, and regularity for stochastic Volterra processes, then bounds differences between processes with different kernels, including pointwise, increment, and Hölder-norm estimates. For convolution kernels, it also provides a path-space bound in Lp(Ω; Lq(0,T)), q ∈ [1,p], and uses kernel-approximation controls to establish convergence in Hölder norms under stated assumptions.

3.1. Existence, uniqueness, and regularity.

For general Volterra kernels, a unique continuous process exists, is stable in C^α([0,T]; Lp(Ω)) for α ∈ (1/p,1], and has ν-Hölder sample paths for ν < α − 1/p.The process is unique up to indistinguishability.Source: §3.1 · pp. 6–7

3.2. Pointwise differences of Volterra processes.

For non-convolution kernels, the difference between a process and its approximation is bounded in C^0([0,T]; Lp(Ω)), extending prior results beyond completely monotone convolution kernels and p = 2.The approximation family is assumed to share the process’s initial value and coefficients, with kernel assumptions using the same regularity constants.Source: §3.2 · p. 10
For convolution kernels, the path difference is bounded in Lp(Ω; Lq(0,T)) for q ∈ [1,p], with a constant independent of q and the approximation index.This estimate follows by combining the pointwise bound with Minkowski’s integral inequality.Source: §3.2 · p. 12

3.3. Differences of Volterra processes in H¨older norms.

The difference is treated in C^ν([0,T]; Lp(Ω)) and Lp(Ω; C^ν([0,T])) for exponents up to the maximal Hölder regularity permitted by the processes.The increment estimate depends only on the kernel differences.Source: §3.3 · pp. 12–13
In Hölder norms, increments of the difference process are bounded using kernel-difference terms over [s,t] and increments of those differences over [0,s], in L1 for the drift kernel and L2 for the diffusion kernel.The bound holds uniformly over the approximation family under the stated shared regularity assumptions.Source: §3.3 · p. 13

3.3. Differences of Volterra processes in H¨older norms

Corollary 3.12 bounds process increments in L^p using the L^1 and L^2 norms of the corresponding increments of the kernel differences, under α_T > 1/p.It applies to convolution kernels and approximations with common coefficients and initial value.Source: §3.3 · p. 15
Although each kernel is monotone, their differences need not be, so Lemma 3.14 imposes additional controls on kernel-difference increments to bound the remaining terms.The subsequent convergence result combines these controls with the L^1 and L^2 approximation accuracy measured by θ_T(ξ).Source: §3.3 · pp. 16–17
Under Assumption 3.13, Theorem 3.15 bounds approximation error uniformly in ξ; if β > 1/p, the difference has pathwise ν-Hölder control for ν ∈ (0, (α ∧ β) − 1/p).The bound also includes C^0([0,T]; L^p(Ω)) accuracy proportional to θ_T(ξ), the kernel-approximation accuracy measure.Source: §3.3 · p. 17

4. Sinc quadrature approximations for the fractional kernel

The section approximates the fractional kernel using sinc quadrature after expressing it through the gamma function and changing variables to obtain an integral over the real line. It establishes pointwise and truncated-series error bounds, including bounds in Lebesgue spaces and for derivatives.

4. Sinc quadrature approximations for the fractional kernel

A gamma-function representation and logarithmic change of variables turn the fractional kernel into an integral over the real line, which is approximated by sinc quadrature.The sinc rule is a trapezoidal rule whose error decays exponentially in the reciprocal quadrature step size.Source: §4 · p. 18

4.1. Pointwise error of the infinite sinc series.

For each fixed t, the infinite sinc series approximates t^-γ with error of order e^-ad/h when the integrand belongs to the required Hardy space on a complex strip.The paper verifies this Hardy-space condition for the fractional-kernel integrand for strip half-widths in (0, π/2).Source: §4.1 · pp. 18–19

4.2. Error of the truncated series in Lebesgue spaces.

Theorem 4.3 bounds truncated-series errors in Lr(s,t), with distinct interval conditions, and also bounds derivative errors for s>0.For intervals beginning at zero, the lemma requires γr<1; the theorem's constants are independent of h and the truncation parameters.Source: §4.2 · p. 21
Calibrating the truncation errors to the sinc-quadrature error permits the resulting error bound to be expressed in terms of the total node count N.The stated node count is N=M−+M++1.Source: §4.2 · p. 23

5. Markovian approximations for stochastic Volterra equations

This section develops sinc-based Markovian approximations for stochastic Volterra processes with fractional convolution kernels and establishes strong-convergence bounds in pointwise, pathwise, and Hölder norms. It specifies the kernel conditions for the Hölder analysis and expresses the fractional-kernel error estimate in terms of quadrature node counts.

5. Markovian approximations for stochastic Volterra equations.

The section applies sinc quadrature error estimates to earlier Volterra-process bounds to obtain strong convergence, including pathwise Hölder convergence.The section overview identifies these as estimates in various norms; the supplied passages here establish the kernel conditions for the Hölder extension.Source: §5 · p. 24, §5.3 · p. 28

5.1. Sinc-Markovian approximations for the fractional kernel.

Replacing a completely monotone kernel’s integral representation by a finite exponential sum yields a Volterra approximation represented by a finite-dimensional Markov process.For fractional convolution kernels, sinc quadrature supplies the exponential-sum kernels and induces corresponding sinc-Markovian approximations.Source: §5.1 · pp. 24–25

5.2. Strong convergence in pointwise and pathwise Lebesgue sense.

Sinc-based Markovian approximations converge strongly for fractional-kernel Volterra processes in pointwise C^0([0,T];L^p(Ω)) and pathwise L^p(Ω;L^q(0,T)) norms.The pathwise result covers q ∈ [1,p]; the error bound can also be expressed in terms of the quadrature-node counts.Source: §5.2 · pp. 27–28

5.3. Strong convergence in H¨older norms.

The fractional-kernel sinc approximations satisfy the stricter kernel conditions required for the paper’s pathwise Hölder error analysis.The proposition specifies an exponentially small kernel-error term θ_T(h) = 3c_T e^(-(π^2−δ)/h), with constants independent of h.Source: §5.3 · p. 28
The bound stated immediately before the excerpt has a constant independent of h for every T > 0 and h ∈ (0, h0].The excerpt omits the bound’s left-hand side and exact form.Source: §5.3 · p. 29
For β⋆ > 1/p, the stated Hölder bound applies for every ν in (0, β⋆ − 1/p), with a constant independent of h.The passage specifies this range for all h ∈ (0, h0], but the bound itself is omitted from the supplied excerpt.Source: §5.3 · p. 29
The fractional-kernel error estimate is expressed using N1 and N2, the quadrature node counts for approximating k1 and k2, respectively.The excerpt names these counts but truncates the resulting error expression.Source: §5.3 · p. 29
Taking β⋆ down to 0 recovers the C0([0, T]; Lp(Ω)) result, but the pointwise and Hölder bounds are discussed separately because the integrability criterion requires β⋆ > 0.The criterion is Assumption 3.13(ii).Source: §5.3 · p. 29

6. Numerical experiments

The experiments empirically verify sinc quadrature errors for the fractional kernel and strong errors for its Markovian approximations. They compare theoretical rate factors and examine error behavior across Hurst parameters and regularity settings.

6. Numerical experiments

The sinc-node counts are calibrated to scale roughly as n, and the strong-error simulations use covariance Cholesky factors for the fractional Brownian motion and its approximation.The simulations use the Riemann–Liouville fractional Brownian motion with varying Hurst parameter H.Source: §6 · p. 30, §6.2 · p. 33

6.1. Sinc quadrature errors.

For r=1, sinc rate factors intersect GG and NGG factors at H≈0.18 and H≈−0.07; for r=2, intersections with Gaussian schemes occur at H≈0.46 and H≈0.37.At Hurst parameters at or below these thresholds, the sinc factor outperforms those Gaussian quadrature factors, particularly in (super) rough regimes.Source: §6.1 · p. 30, Table 1 · p. 30, Figure 1 · p. 31
Figure 3 and Table 3 verify Hölder regularity of t↦||Δ||L^r(0,t) for r∈{1,2} across several Hurst parameters.The experiments examine sup_{0≤s<t≤1}(t−s)^−β||Δ||L^r(s,t), an upper bound for the mapping's Hölder seminorm.Source: §6.1 · p. 32, Table 3 · p. 32

6.2. Strong stochastic errors.

Observed log-squared strong-error behavior confirms the strong error bounds of Theorems 5.7 and 5.11.The experiments consider C^β([0,1]; L^2(Ω)) and L^2(Ω; C^β([0,1])) for β ∈ [0,H).Source: §6.2 · pp. 33–34

Appendix A. Volterra kernels and a Gr¨onwall type inequality

The appendix introduces Volterra kernels and their relevant properties, then uses resolvent equations to establish a generalized Grönwall-type inequality for non-convolution kernels.

Appendix A. Volterra kernels and a Gr¨onwall type inequality

The appendix introduces Volterra kernels and their properties, then establishes a generalized Grönwall-type inequality for non-convolution kernels using resolvent equations.Source: Appendix A · p. 35

A.1. Auxiliary results on Volterra kernels.

This subsection sets out auxiliary Volterra-kernel results, including continuity properties and conditions under which convolution-kernel assumptions transfer to associated non-convolution kernels.

A.1. Auxiliary results on Volterra kernels.

A Volterra kernel is a Borel measurable function that vanishes almost everywhere outside the set where s≤t.Source: §A.1 · p. 35
Under the stated assumptions, each kernel mapping t↦K_j(t,·) has a unique representative in C^α([0,T];L^j(0,T)).The proof extends the mapping continuously from a dense, full-measure subset of [0,T].Source: §A.1 · p. 36
Uniform properties of non-negative, non-increasing convolution kernels imply that the associated non-convolution kernels satisfy Assumption 3.4 with the same constants.The auxiliary estimates bound increments of non-negative, non-increasing convolutions in L^q norms.Source: §A.1 · pp. 37–38

A.2. Resolvent equations and Gr¨onwall’s inequality.

This subsection establishes existence and a series representation for resolvents under stated kernel conditions, then uses them to obtain a Grönwall-type inequality.

A.2. Resolvent equations and Gr¨onwall’s inequality.

Under the stated conditions, a Volterra kernel has a unique resolvent represented by an absolutely convergent Neumann series.The series consists of iterated kernel products.Source: §A.2 · p. 38
When κ is non-negative almost everywhere, its Neumann series and resolvent are also non-negative almost everywhere.Source: §A.2 · p. 38
The resulting Grönwall inequality bounds x(t) by a constant times the running supremum of f on [0,t].The constant depends only on κ and T.Source: §A.2 · p. 39

Appendix B. Auxiliary inequalities

Appendix B provides a Burkholder–Davis–Gundy-type inequality for stochastic Volterra processes and an auxiliary maximization result used with the fractional kernel. The inequality assumes regularity of the kernel and integrability of the stochastic integrand, with a constant growing at most linearly in p.

Appendix B. Auxiliary inequalities

The tailored Burkholder–Davis–Gundy inequality applies under stated continuity and integrability assumptions on the kernel and stochastic integrand.It is formulated for progressively measurable processes and p ≥ 2.Source: Appendix B · p. 39
The auxiliary maximization lemma identifies a unique maximum for f(x) = x^εe^−sx, used to bound the fractional-kernel integrand in the paper's integral representation.The lemma is frequently used in Sections 4 and 5.Source: Appendix B · pp. 39–40
The inequality's positive constant Bp depends only on p and satisfies Bp = O(p).Source: Appendix B · p. 40

Funding

The work of KK was funded by an ERC grant from the European Union; the authors disclaim EU and ERC endorsement or responsibility for their views.

Funding

KK's work was funded by the European Union through ERC grant FunCalc4Stats, agreement no. 101219883.Source: Funding · p. 40
The authors state that their views and opinions do not necessarily reflect those of the European Union or European Research Council.Source: Funding · p. 40
The European Union and the granting authority disclaim responsibility for the authors' views and opinions.Source: Funding · p. 40