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

摘要

Let X-t = integral(t)(0) u(s) dW(s) be the indefinite Skorohod integral on Wiener space (Omega, H, P), and let L-t (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 omega, is in the fractional Sobolev spaces D-alpha,D-p (alpha < 1/2 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 is an element of (0, 1/2). (C) 2006 Elsevier Inc. All rights reserved.