Gaussian fluctuations for stochastic Volterra equations with small noise

TL;DR

The paper investigates Gaussian fluctuations of stochastic Volterra equations with small noise by establishing a quantitative central limit behavior for the fluctuation $\tilde{X}_{\varepsilon,t}$ around its deterministic limit $x_t$. Using Malliavin calculus, it derives an explicit total-variation bound between $\tilde{X}_{\varepsilon,t}$ and its Gaussian limit $Y_t$, and proves a second-order expansion showing $(\tilde{X}_{\varepsilon,t}-Y_t)/\varepsilon$ converges to $(1/2)Z_t$, yielding a limit formula in terms of $\delta(Z_t D Y_t)$. The results are applied to stochastic Volterra equations with a fractional Brownian motion kernel $K_H$, yielding explicit $t$-dependent bounds that depend on the Hurst parameter $H$, and verifying the assumptions for these kernels. This work provides quantitative Gaussian fluctuation theory for a broad class of Volterra equations in small-noise regimes and connects Malliavin-calculus techniques with precise asymptotics in both standard Brownian and fractional Brownian kernel settings.

Abstract

In this paper, we consider a general class of stochastic Volterra equations with small noise. Our aim is to study the fluctuation of the solution around its deterministic limit. We use the techniques of Malliavin calculus to show that the fluctuation process satisfies central limit theorem and provide an optimal estimate for the rate of convergence. An application to stochastic Volterra equations with fractional Brownian motion kernel is given to illustrate the theory.

Theorems & Definitions (20)

• Lemma 2.1
• proof
• Theorem 3.1
• Theorem 3.2
• Remark 3.1
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3