A Generalized Occupation Time Formula For Continuous Semimartingales

Abstract

We show that for a wide class of functions $F$ that: $$ {\lim_{\epsilon \downarrow 0} {\frac{1}{\epsilon}} \int_0^t \Big\{F(s, X_s) - F(s, X_s - \epsilon)\Big\} d\big<X,X\big>_s} = - \int_0^t\int_{\R} F(s, x) d L_s^x $$ where $X_t$ is a continuous semi-martingale, $(L_t^x, x \in \R, t \geq 0)$ its local time process and $(\big<X,X\big>_t, t \geq 0)$ its quadratic variation process.