On Riemann-Liouville type operators, BMO, gradient estimates in the L\'evy-It\^o space, and approximation

TL;DR

This paper explores the relationships between Riemann-Liouville operators, BMO spaces, and approximation of stochastic integrals within the Lévy-Itô framework, providing bounds and decompositions for gradient processes in stochastic analysis.

Contribution

It introduces new bounds for gradient processes on Lévy-Itô spaces and applies these to approximation and decomposition problems for Hölder functionals.

Findings

Upper bounds for gradient processes using BMO conditions

Lower bounds based on oscillatory quantities

Orthogonal decomposition of Hölder functionals into stochastic integrals

Abstract

We discuss in a stochastic framework the interplay between Riemann-Liouville type operators applied to stochastic processes, real interpolation, bounded mean oscillation, and an approximation problem for stochastic integrals. We provide upper and lower bounds for gradient processes on the L\'evy-It\^o space, which arise in the special case of the Wiener space from the Feynman-Kac theory for parabolic PDEs. The upper bounds are formulated in terms of BMO-conditions on the fractional integrated gradient, the lower bounds in terms of oscillatory quantities. On the general L\'evy-It\^o space we are concerned with gradient processes with values in a Hilbert space, where the regularity depends on the direction in this Hilbert space. We discuss two applications of our techniques: on the Wiener space an approximation problem for H\"older functionals and on the L\'evy-It\^o space an orthogonal decomposition of H\"older functionals into a sum of stochastic integrals with a control of the corresponding integrands.