Fractional smoothness for the generalized local time of the indefinite Skorohod integral

AbstractLet Xt=∫0tusdWs be the indefinite Skorohod integral on Wiener space (Ω,H,P), and let Lt(x) be its the generalized local time introduced by Tudor in [C.A. Tudor, Martingale-type stochastic calculus for anticipating integral processes, Bernoulli 10 (2004) 313–325]. We prove that the generalized local time, as a nonlinear functional of ω, is in the fractional Sobolev spaces Dα,p (α<12 and p>2) under some conditions imposed on the anticipating integrand u via the technique of Malliavin calculus and the K-method in the real interpolation theory. The result is optimal for the fractional Brownian motion with the Hurst parameter h∈(0,12)