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Quelques approximations du temps local brownien

Blandine Berard Bergery 1 Pierre Vallois 1, 2
2 OMEGA - Probabilistic numerical methods
CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, CNRS - Centre National de la Recherche Scientifique : UMR7502
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}{\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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Submitted on : Wednesday, April 25, 2007 - 5:27:15 PM
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Blandine Berard Bergery, Pierre Vallois. Quelques approximations du temps local brownien. Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2007, 345, pp.45-48. ⟨10.1016/j.crma.2007.05.007⟩. ⟨hal-00098326v3⟩

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