A Malliavin-Skorohod calculus in \(L^0\) and \(L^1\) for additive and Volterra-type processes

Abstract

In this paper we develop a Malliavin-Skorohod type calculus for additive processes in the \(L^0\) and \(L^1\) settings, extending the probabilistic interpretation of the Malliavin-Skorohod operators to this context. We prove calculus rules and obtain a generalization of the Clark-Hausmann-Ocone formula for random variables in \(L^1\). Our theory is then applied to extend the stochastic integration with respect to volatility modulated L\'evy-driven Volterra processes recently introduced in the literature. Our work yields to substantially weaker conditions that permit to cover integration with respect, e.g. to Volterra processes driven by \(\alpha\)-stable processes with \(\alpha < 2\). The presentation focuses on jump type processes.