Journal of Applied Mathematics and Stochastic Analysis 
Volume 2006 (2006), Article ID 26961, 5 pages
doi:10.1155/JAMSA/2006/26961

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

Abstract

Let B=(Bt)t≥0 be a standard Brownian motion and let (Ltx;t≥0,x∈ℝ) be a continuous version of its local time process. We show that the following limitlim⁡ε↓0(1/2ε)∫0t{F(s,Bs−ε)−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.