Weak rough kernel comparison via PPDEs for integrated Volterra processes

TL;DR

This paper analyzes the impact of kernel approximation on integrated Volterra processes with small Hurst parameter, using PPDEs to quantify weak errors in terms of kernel differences, relevant to physics and finance applications.

Contribution

It introduces a novel PPDE-based framework to quantify weak errors caused by kernel approximation in integrated Volterra processes with small Hurst parameter.

Findings

Quantifies weak error using L1 norms of kernel differences.

Provides a path-dependent Feynman-Kac formula for analysis.

Framework applicable to physics and financial models.

Abstract

Motivated by applications in physics (e.g., turbulence intermittency) and financial mathematics (e.g., rough volatility), this paper examines a family of integrated stochastic Volterra processes characterized by a small Hurst parameter $H<\tfrac{1}{2}$. We investigate the impact of kernel approximation on the integrated process by examining the resulting weak error. Our findings quantify this error in terms of the $L^1$ norm of the difference between the two kernels, as well as the $L^1$ norm of the difference of the squares of these kernels. Our analysis is based on a path-dependent Feynman-Kac formula and the associated partial differential equation (PPDE), providing a robust and extendible framework for our analysis.