Limit distribution of errors in discretization of stochastic Volterra equations with multidimensional kernel

Abstract: This paper investigates the limit distribution of discretization errors in stochastic Volterra equations (SVEs) with general multidimensional kernel structures. While prior studies, such as Fukasawa and Ugai (2023), were focused on one-dimensional fractional kernels, this research generalizes to broader classes, accommodating diagonal matrix kernels that include forms beyond fractional type. The main result demonstrates the stable convergence in law for the rescaled discretization error process, and the limit process is characterized under relaxed assumptions.

• The paper establishes stable convergence in law for the rescaled discretization error in multidimensional kernel stochastic Volterra equations.
• It introduces novel operator-theoretic techniques under relaxed assumptions to derive explicit convergence rates and a Gaussian error term.
• The results offer robust foundations for developing numerical schemes with quantifiable bias and variance in applications like rough volatility modeling.

Limit Laws for Discretization Errors in Multidimensional-Kernel Stochastic Volterra Equations

Introduction

The study provides a rigorous theoretical analysis of discretization errors in stochastic Volterra equations (SVEs) featuring multidimensional kernel structures. Classic work on SDE discretization error distributions has been extended to some SVEs, but primarily to those with scalar, fractional kernels. This paper closes a significant gap by treating SVEs with general diagonal matrix kernels, encapsulating a wider variety of non-Markovian dynamics relevant for fields such as mathematical finance, quantitative biology, and signal processing. The results are not limited to conventional fractional (e.g., $u^{H-1/2}$) kernels but encompass matrix-valued and non-fractional forms, importantly accommodating locally-stochastic rough volatility modeling scenarios.

Mathematical Framework

Consider the SVE

$$X_t = X_0 + \int_0^t b(X_s) \, ds + \int_0^t \varphi(t-s) \sigma(X_s) \, dW_s,$$

on a filtered probability space, where the kernel $\varphi$ has diagonal structure $$\varphi(t) = \mathrm{diag}(\varphi_1(t),\dots,\varphi_d(t))$$, covering a multidimensional setting, and $b$ and $\sigma$ are sufficiently regular. Crucially, the analysis is performed under quite general structural and regularity conditions (A-$(H, a, c_1, ..., c_d)$) on the $\varphi_i$, far beyond the fractional regime.

Instead of classical Euler-Maruyama discretization, the paper considers a natural stepwise approximation $X^n$, and the primary object of study is the rescaled discretization error

$U^n = n^H(X - X^n)$

for $H\in (0,1)$ determined by the asymptotics of the kernel.

Main Theoretical Results

The key result establishes stable convergence in law of $U^n$ (viewed as a random element of a Hölder space) under broad kernel conditions:

• $U^n$ converges stably in $C^{1/2-a-\varepsilon}([0,T],\mathbb{R}^d)$ for any admissible $a$ and $\varepsilon$.
• The limit process $U$ is the unique strong solution to a "limit SVE," with leading-order drift and diffusion terms inherited from the original equation's coefficients and kernel, plus an explicit, Gaussian error term reflecting the discretization's influence.

This limit SVE incorporates multiple stochastic integrals driven by an augmented Brownian motion (independent of the original system), reflecting the non-trivial coupling and error accumulation induced by discretization under non-Markovian and non-scalar kernels. The normalization exponent $H$ is shown to characterize the critical discretization rate via explicit kernel scaling asymptotics.

A highlight is that all proofs, including tightness, moment bounds, and limit identification, are achieved under relaxed regularity and structural assumptions. This robustness is underpinned by intricate operator-theoretic estimates on Hölder and Young convolution spaces, direct analysis of the diagonally-kernelized discretization error decomposition, and adaptation of invariance principles for random elements in nonseparable function spaces.

Numerical and Analytical Implications

Sharp convergence rates for the discretization error in $L^p$ and Hölder norms are derived, generalizing classical SDE results and those for fractional SVEs. The law of the normalized error process is rendered explicit, giving practitioners precise quantification of bias and randomness in strong approximation schemes for multidimensional-kernel SVEs.

Moreover, the analysis reveals that discretization error distributions are essentially universal within the kernel class handled: the limiting error law depends only on the leading-order scaling of the kernel and certain coefficient combinations. When applied to local-stochastic rough volatility models and other non-scalar-kernel SVEs, this insight provides a rigorous foundation for constructing and analyzing numerical methods with quantifiable statistical uncertainty.

Theoretical and Future Directions

By moving beyond the fractional, scalar-kernel paradigm, the work opens the way to:

• Studying the impact of kernel structure (including off-diagonal matrix elements or more exotic forms) on discretization error propagation.
• Developing higher-order or adaptively-weighted schemes using the explicit asymptotic error characterization.
• Rigorous error analysis for SVE-driven SPDEs or path-dependent control problems where multidimensional kernel effects are non-negligible.
• Investigating the interplay between kernel singularity and solution path-regularity for both theoretical and algorithmic advances.

The generality of the conditions imposed (especially on the kernel's leading singularity and regularity) suggests possible direct extensions to operator-valued, possibly non-diagonal kernels, provided suitable moment and continuity controls are available.

Conclusion

This study rigorously characterizes the limiting distribution of discretization errors for SVEs with broad classes of multidimensional diagonal kernels. By establishing stable convergence laws and obtaining explicit rate and law results under minimal conditions, it provides a comprehensive foundation for robust numerical analysis and practical simulation of non-Markovian SVEs beyond the fractional framework. The techniques and results have broad applicability, pointing toward future research on even more general kernel structures and their impact on stochastic integration in high-dimensional or non-Markovian settings.