A Generalized Occupation Time Formula For Continuous Semimartingales

Abstract

We show that for a wide class of functions $F$ that: $$\underset{\epsilon \downarrow 0}{\lim }\frac{1}{\epsilon }{\int }_0^t\{F(s,X_s)-F(s,X_s-\epsilon )\}d⟨X,X{⟩}_s=-{\int }_0^t{\int }_{\mathbb{R}}F(s,x)d{L}_s^x$$ where $X_t$ is a continuous semi-martingale, $(L_t^x,x\in \mathbb{R},t\ge 0)$ its local time process and $(⟨X,X{⟩}_t,t\ge 0)$ its quadratic variation process.

3 pages