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

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.