Quelques approximations du temps local brownien

Abstract

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}ε\int_0^{t} X_{(u+ε)\wedge t}^+ \indi_{\{X_u \leqslant 0\}} du + \frac{2}ε\int_0^{t} X_{(u+ε) \wedge t}^- \indi_{\{X_u>0\}} du$ and $\frac{4}ε\int_0^{t} X_u^- \indi_{\{X_{(u+ε) \wedge t} > 0\}} du$ converge in the ucp sense to $L_t^0$, as $ε\to 0$. We show that $ \frac{1}ε\int_0^t (\indi_{\{x<X_{s+ε}\}} - \indi_{\{x<X_{s}\}}) (X_{s+ε}-X_{s})ds$ goes to $L_t^x$ in $L^2(Ω)$ as $ε\to 0$, and that the rate of convergence is of order $ε^α$, for any $α< {1/4}$.