Quelques approximations du temps local brownien

Mathematics – Probability

Scientific paper

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Soumis dans les Comptes rendus - Math\'ematique

Scientific paper

We give some approximations of the local time process $(L_t^x)_{t\geqslant 0}$ at level $x$ of the real Brownian motion $(X_t)$. We prove that $ \frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon)\wedge t}^+ \indi_{\{X_u \leqslant 0\}} du + \frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon) \wedge t}^- \indi_{\{X_u>0\}} du$ and $\frac{4}{\epsilon}\int_0^{t} X_u^- \indi_{\{X_{(u+\epsilon) \wedge t} > 0\}} du$ converge in the ucp sense to $L_t^0$, as $\epsilon \to 0$. We show that $ \frac{1}{\epsilon}\int_0^t (\indi_{\{x

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