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

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.