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On the Analysis of a Singular Stochastic Volterra Differential Equation driven by a Wiener Noise

Emmanuel Coffie, Olivier Menoukeu-Pamen, Frank Proske

TL;DR

This work addresses singular stochastic Volterra differential equations driven by Brownian motion, establishing well-posedness for drift fields with a representation $b(t,s,x)=\sum_{m\ge0}(t-s)^m g_m(s,x)$ and bounded components $g_m$. The authors construct a unique strong solution on a short time horizon, prove Malliavin differentiability, and show local Sobolev differentiability of the flow with respect to the initial data. The approach combines weak convergence and approximation by smoother drifts with Malliavin calculus and Girsanov's theorem to bridge weak and strong solutions, supported by compactness and Mitoma-type results. Overall, the paper extends SVDE theory to singular kernels and non-Lipschitz drifts, providing a rigorous framework for non-Markovian dynamics with potential applications in physics, biology, and finance.

Abstract

In this article, we construct unique strong solutions to a class of stochastic Volterra differential equations driven by a singular drift vector field and a Wiener noise. Further, we examine the Sobolev differentiability of the strong solution with respect to its initial value.

On the Analysis of a Singular Stochastic Volterra Differential Equation driven by a Wiener Noise

TL;DR

This work addresses singular stochastic Volterra differential equations driven by Brownian motion, establishing well-posedness for drift fields with a representation and bounded components . The authors construct a unique strong solution on a short time horizon, prove Malliavin differentiability, and show local Sobolev differentiability of the flow with respect to the initial data. The approach combines weak convergence and approximation by smoother drifts with Malliavin calculus and Girsanov's theorem to bridge weak and strong solutions, supported by compactness and Mitoma-type results. Overall, the paper extends SVDE theory to singular kernels and non-Lipschitz drifts, providing a rigorous framework for non-Markovian dynamics with potential applications in physics, biology, and finance.

Abstract

In this article, we construct unique strong solutions to a class of stochastic Volterra differential equations driven by a singular drift vector field and a Wiener noise. Further, we examine the Sobolev differentiability of the strong solution with respect to its initial value.
Paper Structure (4 sections, 8 theorems, 61 equations)

This paper contains 4 sections, 8 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Strong uniqueness
  3. Weak convergence of approximating solutions
  4. Existence of a unique weak solution and proof of the main result

Key Result

Theorem 1

Suppose that $b\in$$\mathbb{H}$. Then there exists for a sufficiently small time horizon $T$ a unique strong solution $X_{t}^{x},0\leq t\leq T$ to the SVDESDE0 for all initial values $x$. Moreover, $X_{t}^{x}$ is Malliavin differentiable and the associated stochastic flow $X_{t}^{\cdot }$ belongs to

Theorems & Definitions (15)

  • Theorem 1
  • Proposition 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark 1
  • Lemma 5
  • proof
  • proof : Proof of Theorem \ref{['Main']}
  • ...and 5 more