Failure of the Markov property for stochastic Volterra equations

TL;DR

This paper demonstrates that stochastic Volterra equations inherently lack the Markov property due to their memory effects, and their Markovian lifts are typically infinite-dimensional, highlighting the need for advanced analytical tools.

Contribution

It proves the failure of the Markov property for general SVEs with H"older-continuous coefficients and broad classes of kernels, extending previous Gaussian process results.

Findings

SVEs do not possess the Markov property.

Markovian lifts of SVEs are generally infinite-dimensional.

Memory effects in SVEs are intrinsically infinite-dimensional.

Abstract

Memory-driven stochastic dynamics arise naturally in many applications, and stochastic Volterra equations (SVEs) offer a flexible framework for modeling such systems. Their convolution structure with Volterra kernels endows the dynamics with a formal path-dependency, which suggests the failure of the Markov property. While this has previously been rigorously established only for Gaussian Volterra processes, by constructing nondegenerate admissible perturbations through Markovian lifts, we prove that also general SVEs with H\"older-continuous coefficients do not possess the Markov property for a broad class of Volterra kernels. Moreover, we show that the associated Markovian lift is, in general, necessarily infinite-dimensional. These observations reflect the intrinsic infinite-dimensionality of memory effects in SVEs and underscore the need for analytical and probabilistic tools beyond the classical Markovian framework.