Un processus ponctuel associé aux maxima locaux du mouvement brownien

Details Un processus ponctuel associé aux maxima locaux du mouvement brownien Un processus ponctuel associé aux maxima locaux du mouvement brownien

2010-04-30

arxiv.org/abs/1004.5530v1

Mathematics

Probability

20 pages, 2 figures.

Scientific paper

10.1007/s00440-009-0236-4

Let $B = (B_t)_{t \in {\bf R}}$ be a symmetric Brownian motion, i.e. $(B_t)_{t \in {\bf R}_+}$ and $(B_{-t})_{t \in {\bf R}_+}$ are independent Brownian motions starting at $0$. Given $a \ge b>0$, we describe the law of the random set $${\cal M}_{a,b} = \{t \in {\bf R} : B_t = \max_{s \in [t-a,t+b]} B_s\},$$ and we describe the L\'evy measure of a subordinator whose closed range is the regenerative set $${\cal R}_a = \{t \in {\bf R}\_+ : B_t = \max_{s \in [(t-a)_+,t]} B_s\}.$$

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