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

TL;DR

This paper demonstrates that stochastic Volterra equations generally lack the time-homogeneous Markov property due to their path-dependent structure, with a specific exception for exponential kernels, supported by new computational methods.

Contribution

The authors develop novel computational techniques based on small-time asymptotics to rigorously prove the failure of the Markov property in SVEs, except for a specific exponential kernel case.

Findings

SVEs generally do not have the time-homogeneous Markov property.

The Markov property holds only for exponential kernels in affine drift cases.

New computational methods are introduced for analyzing SVEs with H"older coefficients.

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 H\"older 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 $K(t) = c e^{-\lambda t}$, where the parameter $\lambda$ is linked to the initial curve of the SVE.