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I'm currently studying Local times of Semimartingales and I came across the following formula for the local time at $a$ of $X \in S_c$: $$ L_t^a (X) = \lim_{\varepsilon \to 0} \frac{1}{2\varepsilon} \int_0^t \mathbb{1}_{(a-\varepsilon,a+\varepsilon)}(X_s) dX_s $$ However, I'm stuck on the proof (and have looked online and in textbooks but with no success). Here's my attempt thus far:

The occupation formula states that for a non-negative function $\phi: \mathbb{R} \to [0,\infty)$: $$\int_0^t \phi(X_s) d\langle X \rangle_s = \int_{\mathbb{R}} \phi (u) L_t^u du$$ Putting $\phi := (2\varepsilon)^{-1} \mathbb{1}_{(a-\varepsilon,a+\varepsilon)}$ I obtain: $$L_{t,\epsilon}^a := \frac{1}{2\varepsilon}\int_{\mathbb{R}}\mathbb{1}_{(a-\varepsilon,a+\varepsilon)}(u) L_t^u du = \int_0^t \frac{1}{2\varepsilon}\mathbb{1}_{(a-\varepsilon,a+\varepsilon)}(X_s) d\langle X \rangle_s $$ Now here's the kicker. I want to justify taking the limit $\varepsilon \to 0$ inside the integral on the LHS so that $L_{t,\epsilon}^a \to L_t^a (X)$ (using a property of the dirac delta function at a). However, I can't see how to suitably bound the integral on the RHS (unless we have a BM in which case it's not to bad) because I was thinking of using DCT (as I did with the BM case).

Any help would be greatly appreciated.

Curtis.Clement
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1 Answers1

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By the Lebesgue differentiation theorem,

$$\lim_{\varepsilon \rightarrow 0} \frac{1}{2\varepsilon} \int_{\mathbb{R}} 1_{(a-\varepsilon,a+\varepsilon)}L_t^udu = \lim_{\varepsilon \rightarrow 0} \frac{1}{2\varepsilon} \int_{B(a,\varepsilon)} L_t^udu = L_t^a $$ for almost every $a$. Therefore, based on the occupation formula computation you did, $$L_t^a = \lim_{\varepsilon \rightarrow 0} \frac{1}{2\varepsilon} \int_{\mathbb{R}} 1_{(a-\varepsilon,a+\varepsilon)} = \lim_{\varepsilon \rightarrow 0} \frac{1}{2\varepsilon} \int_{0}^t 1_{(a-\varepsilon,a+\varepsilon)}(X_s)d\langle X \rangle_s $$ You don't need to take the limit inside the integral on the right because this is the formula for local time you wanted.

user6247850
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