A Malliavin–Skorohod calculus in L⁰ and L¹ for additive and Volterra-type processes

Abstract

In this paper we develop a Malliavin–Skorohod type calculus for additive processes in the and 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 . 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 to e.g. Volterra processes driven by -stable processes with . The presentation focuses on jump type processes.

Keywords:

• Additive processes
• Lévy processes
• -stable processes
• Volterra processes
• Malliavin–Skorohod calculus

AMS Subject Classifications:

• 60H05
• 60H07

Acknowledgements

This work has been developed under the project Stochastic in Environmental and Financial Economics (SEFE) at the Center for Advanced Study (CAS) at the Norwegian Academy of Science and Letters. The authors thank CAS for the support and the kind hospitality.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by [grant number MEC MTM 2012 31192], [grant number MEC MTM 2013 40782 P].