Comparison principles for stochastic Volterra equations

In this work, we establish a comparison principle for stochastic Volterra equations with respect to the initial condition and the drift $b$ applicable to a wide class of Volterra kernels and input curves $g$ that may be singular at zero. The latter appear, e.g., in the study of Markovian lifts for such Volterra equations. For completely monotone kernels, our result holds without any further restrictions, while for not completely monotone kernels, it is shown that such a principle fails unless the drift is additionally monotone. As a side-product of our results, we also complement the literature on the weak existence of continuous nonnegative solutions, which covers the rough Cox-Ingersoll-Ross process with singular initial conditions.