On fractional derivatives of the local time of a symmetric stable process as a doubly indexed process

Abstract

In this paper we prove two main results. The rst one is to prove the regularity of fractional derivatives of local time of symmetric stable process with index 1 < ≤ 2. Our result is similar to that of Marcus and Rosen in 1992 for local time. The second result is to give a ( , )-variation of fractional derivatives of local time of symmetric stable process with index 1 < ≤ 2. Our approach is similar to that of Eisenbaum in 2000 for local time.

Distributional Itô’s Formula and Regularization of Generalized Wiener Functionals

“…The study of Hölder continuity of local times had been initiated by Trotter [22, inequalities (2.1) and (2.3)], in which the almost-sure Hölder-continuity of the Brownian local time {l(t, x) : t 0, x ∈ R} in time-space variable (t, x) was proved (see also Boufoussi-Roynette [6]). There are a lot of such studies (see, e.g., Liang [15], Ait Ouahra-Kissami-Ouahhabi [3], Shuwen-Cheng [19] and references therein). Theorem 1.2 implies immediately the following.…”