A note on Malliavin fractional smoothness for L\'evy processes and approximation

TL;DR

This paper investigates the relationship between approximation rates of stochastic integrals driven by Lévy processes and their Malliavin fractional smoothness, using Besov spaces and real interpolation methods.

Contribution

It establishes a link between L2-approximation rates of stochastic integrals and their Malliavin fractional smoothness for Lévy processes, providing new insights into their regularity properties.

Findings

Approximation rates are characterized by Besov space smoothness.

The approach applies to stochastic exponentials and Lévy martingales.

Results enhance understanding of stochastic integral approximation accuracy.

Abstract

Assume a L\'evy process $X$ on the time interval $[0,1]$ that is an $L_2$-martingale and let $Y$ be either its stochastic exponential or $X$ itself. We consider Riemann-approximations of certain stochastic integrals driven by $Y$ and relate the $L_2$-approximation rates to the Malliavin fractional smoothness of the integral to be approximated. The Malliavin fractional smoothness is described by Besov spaces generated with the real interpolation method.