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

TL;DR

The paper addresses the limit distribution of discretization errors for stochastic Volterra equations with generalized, multidimensional diagonal kernels, extending known results from one-dimensional fractional kernels. It proves that the rescaled error $U^n = n^H (X - \\hat{X})$ converges stably in law to a limit process $U$ that solves a coupled SVE with a kernel-driven drift and diffusion term plus a Brownian-correction term, under the condition A-$(H,\\alpha,c_1,...,c_d)$. The method combines kernel regularity estimates, moment/Hölder bounds for the approximate process, and a careful decomposition of discretization errors into leading and negligible components, following and extending the framework of FukasawaUgai2023, including new results for $H>1/2$. The contributions generalize the kernel structure, yielding a precise limit law for discretization errors and enabling broader non-Markovian modeling (notably rough volatility) through SVEs with diagonal kernels.

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.

Theorems & Definitions (21)

• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Remark 1
• Lemma 8