Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 26961, 5 pages

Some limit theorems connected with Brownian local time

Raouf Ghomrasni

Fakultät II--Mathematik und Naturwissenschaften, Institut für Mathematik, Technische Universität Berlin , Straβe des 17. Juni 136, Berlin 10623, Germany

Received 26 October 2004; Revised 11 April 2005; Accepted 12 April 2005

Copyright © 2006 Raouf Ghomrasni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let B=(Bt)t0 be a standard Brownian motion and let (Ltx;t0,x) be a continuous version of its local time process. We show that the following limitlimε0(1/2ε)0t{F(s,Bsε)F(s,Bs+ε)}ds is well defined for a large class of functions F(t,x), and moreover we connect it with the integration with respect to local time Ltx . We give an illustrative example of the nonlinearity of the integration with respect to local time in the random case.