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 Lévy-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.

27 pages

References in corpus (3)

• Integrability conditions for space-time stochastic integrals: Theory and applications
• An anticipating Itô formula for Lévy processes
• A martingale representation theorem for a class of jump processes