We show that for a wide class of functions
F
we have lim \documentclass{aastex}
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\begin{document}
$$\mathop {\lim }\limits_{\varepsilon \downarrow 0} \frac{1}{\varepsilon }\int\limits_0^t {\{ F(s,X_s ) - F(s,X_s - \varepsilon )\} d\left\langle {X,X} \right\rangle _s = - } \int\limits_0^t {\int\limits_\mathbb{R} {F(s,x)dL_s^x } }$$
\end{document} where
Xt
is a continuous semimartingale, (
Ltx
,
x
∈ ℝ,
t
≧ 0) its local time process and (〈
X, X
〉
t
,
t
≧ 0) its quadratic variation process.
On the quadratic variation process of a continuous Martingale