A Simple Proof of El Karoui's Upcrossing Theorem on Semimartingales

Abstract:

      In the papers [1] and [2] El karoui has given an upcrossing characterigati on of local time for semimartingales, which generalizes partially Lévy's downcrossing theorem on brownian motion. In the present note we propose to deduce this result directly from the dominated convergence theorem of stochastic integrals, so our proof is much simpler than the original one.