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Weak rough kernel comparison via PPDEs for integrated Volterra processes

Mireille Bossy, Kerlyns Martinez, Paul Maurer

TL;DR

The paper addresses weak errors from kernel-approximation in integrated stochastic Volterra processes with fractional-type kernels $K$ and small $H\in(0,\tfrac{1}{2})$, using a path-dependent PDE framework and the functional Itô formula. It derives an explicit $L^1$-type bound for the weak error $|\mathbb{E}[\phi(X_T)]-\mathbb{E}[\phi(\overline{X}_T)]|$ in terms of $\|K-\overline{K}\|_{L^1}$ and $\|K^2-\overline{K}^2\|_{L^1}$, with a constant $C_K$ depending on $\|K\|_{L^2}$ and related norms. The analysis hinges on Hadamard differentiability for path derivatives, a trajectory-dependent Feynman-Kac formula, and a careful decomposition of the error into parts that separately reflect the two kernel-difference terms. The results provide a robust, extendible tool for assessing kernel-approximation schemes (e.g., Markovian Markovian-approximations) in rough volatility and related intermittent phenomena, and quantify how kernel smoothness and singularity affect weak convergence in integrated Volterra 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.

Weak rough kernel comparison via PPDEs for integrated Volterra processes

TL;DR

The paper addresses weak errors from kernel-approximation in integrated stochastic Volterra processes with fractional-type kernels and small , using a path-dependent PDE framework and the functional Itô formula. It derives an explicit -type bound for the weak error in terms of and , with a constant depending on and related norms. The analysis hinges on Hadamard differentiability for path derivatives, a trajectory-dependent Feynman-Kac formula, and a careful decomposition of the error into parts that separately reflect the two kernel-difference terms. The results provide a robust, extendible tool for assessing kernel-approximation schemes (e.g., Markovian Markovian-approximations) in rough volatility and related intermittent phenomena, and quantify how kernel smoothness and singularity affect weak convergence in integrated Volterra 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 . 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 norm of the difference between the two kernels, as well as the 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.
Paper Structure (20 sections, 13 theorems, 157 equations)

This paper contains 20 sections, 13 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Path-dependent Feynman-Kac formula.
  3. Directional derivatives.
  4. Preamble on Volterra processes
  5. Main results
  6. Preliminary tools, functional Itô formula and Path-Dependent PDES
  7. Orthogonal decomposition and flow processes
  8. Regularity of $\omega \mapsto \mathbb{E}[\phi(X_T^{t,x,\omega})]$ -- the case of continuous directions for $\partial_\omega$
  9. 1. Hadamard-differentiability and associated probabilistic representations.
  10. 2. Continuity of the derivatives of $u$ under the distance $d_{\bar{\Lambda}}$.
  11. 3. Growth control of the derivatives.
  12. Regularity of $\omega\mapsto \mathbb{E}[\phi(X_T^{t,x,\omega})]$ -- the case of singular directions for $\partial_\omega$
  13. Path dependant Feynman-Kac formula and Functional Itô formula
  14. Proof of Theorem \ref{['thm:weakcvg_integrated']}
  15. Contributions in $\|K-\overline{K}\|_{L^1([0,T])}$
  16. ...and 5 more sections

Key Result

Lemma 2.1

For a test function $\phi \in C^2(\mathbb{R})$ such that, for all $x\in \mathbb{R}$, $|\phi(x)| + |\phi'(x)| + |\phi"(x)| \lesssim (1+|x|^{\kappa_{\phi}})$, for some $\kappa_{\phi}>0$,

Theorems & Definitions (26)

  • Lemma 2.1
  • Theorem 3.1: Weak comparison
  • Remark 3.2
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.3
  • Lemma 4.4
  • proof
  • Proposition 4.5
  • Remark 4.6
  • ...and 16 more