Itô formula for the two-parameter fractional Brownian motion using the extended divergence operator

Abstract

We develop a stochastic calculus of divergence type with respect to the fractional Brownian sheet (fBs) with any Hurst parameters in and beyond the fractional scale. We define stochastic integration in the extended Skorohod sense, and derive Itô and Tanaka formulas. In the case of Gaussian fields that are more irregular than fBs for any Hurst parameters, we are able to complete the same program for those Gaussian fields that are almost-surely uniformly continuous.