Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than 12

In this paper we introduce a stochastic integral with respect to the process Bt=∫0t(t−s)−α dWs where 0<α<1/2, and Wt is a Brownian motion. Sufficient integrability conditions are deduced using the techniques of the Malliavin calculus and the notion of fractional derivative. We study continuity properties of the indefinite integral and we derive a maximal inequality.