On Riemann–Liouville type operators, bounded mean oscillation, gradient estimates and approximation on the Wiener space

We discuss in a stochastic framework the interplay between Riemann–Liouville type operators applied to stochastic processes, bounded mean oscillation, real interpolation, and approximation. In particular, we investigate the singularity of gradient processes on the Wiener space arising from parabolic PDEs via the Feynman–Kac theory. The singularity is measured in terms of bmo-conditions on the fractional integrated gradient. As an application we treat an approximation problem for stochastic integrals on the Wiener space. In particular, we provide a discrete time hedging strategy for the binary option with a uniform local control of the hedging error under a shortfall constraint.