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

TL;DR

This paper extends Malliavin-Skorohod calculus to additive and Volterra-type processes in $L^0$ and $L^1$, enabling stochastic integration under weaker conditions, especially for jump processes like stable Lévy processes.

Contribution

It develops a Malliavin-Skorohod calculus framework in $L^0$ and $L^1$, generalizes the Clark-Hausmann-Ocone formula, and broadens stochastic integration to include stable Lévy-driven Volterra processes.

Findings

Extended calculus rules for additive processes.

Generalized Clark-Hausmann-Ocone formula in $L^1$.

Applicable to stable Lévy processes with $eta<2$.

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.