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Stochastic Volterra equations: failure of the time-homogeneous Markov property

Martin Friesen, Stefan Gerhold, Kristof Wiedermann

TL;DR

The paper proves that stochastic Volterra equations generally cannot possess the time-homogeneous Markov property due to path-dependence encoded by the Volterra kernel. It develops two complementary routes: a moment-configuration analysis for affine drifts showing the Markov property forces an exponential kernel, and a small-time CLT reduction to Gaussian limits that yields non-Markovity for a broad class of Hölder-coefficient SVEs. A direct Gaussian-kernel argument confirms non-Markovity for the fractional Riemann–Liouville kernel when $H \neq 1/2$. Together, these results delineate precisely when SVEs can be treated as Markov processes and guide numerical and modeling choices in rough volatility and memory-filled systems.

Abstract

Path-dependence is a defining feature of many real-world systems, with applications ranging from population dynamics to rough volatility models and electricity spot prices. In stochastic Volterra equations (SVEs), such dependence is encoded in the Volterra kernel, which dictates how past trajectories influence present dynamics on infinitesimal time scales. This structure suggests a breakdown of the Markov property. In this article, we develop computational techniques and methods based on small-time asymptotics for SVEs with Hölder coefficients to rigorously establish that they cannot possess the time-homogeneous Markov property. In particular, for affine drifts, we prove that the time-homogeneous Markov property only holds in the case of the exponential Volterra kernel $K(t) = c e^{-λt}$, where the parameter $λ$ is linked to the initial curve of the SVE.

Stochastic Volterra equations: failure of the time-homogeneous Markov property

TL;DR

The paper proves that stochastic Volterra equations generally cannot possess the time-homogeneous Markov property due to path-dependence encoded by the Volterra kernel. It develops two complementary routes: a moment-configuration analysis for affine drifts showing the Markov property forces an exponential kernel, and a small-time CLT reduction to Gaussian limits that yields non-Markovity for a broad class of Hölder-coefficient SVEs. A direct Gaussian-kernel argument confirms non-Markovity for the fractional Riemann–Liouville kernel when . Together, these results delineate precisely when SVEs can be treated as Markov processes and guide numerical and modeling choices in rough volatility and memory-filled systems.

Abstract

Path-dependence is a defining feature of many real-world systems, with applications ranging from population dynamics to rough volatility models and electricity spot prices. In stochastic Volterra equations (SVEs), such dependence is encoded in the Volterra kernel, which dictates how past trajectories influence present dynamics on infinitesimal time scales. This structure suggests a breakdown of the Markov property. In this article, we develop computational techniques and methods based on small-time asymptotics for SVEs with Hölder coefficients to rigorously establish that they cannot possess the time-homogeneous Markov property. In particular, for affine drifts, we prove that the time-homogeneous Markov property only holds in the case of the exponential Volterra kernel , where the parameter is linked to the initial curve of the SVE.
Paper Structure (4 sections, 5 theorems, 85 equations)

This paper contains 4 sections, 5 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Moment formula approach for affine drifts
  3. Gaussian Riemann-Liouville case
  4. Non-Markov property by reduction via a small-time CLT

Key Result

Theorem 2.1

Assume $K\in L_{\mathrm{loc}}^2(\mathbb{R}_+)\cap C(\mathbb{R}_+^*)$, let $D\subseteq \mathbb{R}$ be closed, $\sigma:D \longrightarrow \mathbb{R}$ be continuous with linear growth, and fix $\lambda \in \mathbb{R}$. Suppose that: Then there exists $c \in \mathbb{R}$ such that $K(t) = c\space e^{-\lambda t}$ for each $t \in\mathbb{R}_+$.

Theorems & Definitions (13)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Theorem 4.2
  • ...and 3 more