Convex ordering for stochastic Volterra equations and their Euler schemes

In this paper, we are interested in comparing solutions to stochastic Volterra equations for the convex order on the space of continuous Rd-valued paths and for the monotonic convex order when d=1. Even if these solutions are in general neither semimartingales nor Markov processes, we are able to exhibit conditions on their coefficients enabling the comparison. Our approach consists in first comparing their Euler schemes and then taking the limit as the time step vanishes. We consider two types of Euler schemes, depending on the way the Volterra kernels are discretised. The conditions ensuring the comparison are slightly weaker for the first scheme than for the second, and this is the other way around for convergence. Moreover, we weaken the integrability needed on the starting values in the existence and convergence results in the literature to be able to only assume finite first moments, which is the natural framework for convex ordering.