Sur un calcul de F. Knight. (On a calculus of F. Knight). (French) Zbl 0654.60061

Biane, Ph.

Séminaire de probabilités XXII, Strasbourg/France, Lect. Notes Math. 1321, 190-196 (1988).

[For the entire collection see Zbl 0635.00013.]
Let B be a real Brownian motion starting from 0, \(L_ t\) its local time at 0, and define \[ \tau (\alpha)=\inf \{t>0;\quad L_ t>\alpha \},\quad M_{\alpha}=\sup \{| B_ s|;s\leq \tau (\alpha)\}. \] F. Knight [Inverse local times, positive sojourns and maxima for Brownian motion, to appear in Astérisque] has computed the law of the random variable \(\tau (\alpha)/M^ 2_{\alpha}:\) its Laplace transform is \[ E[\exp (-\lambda)\tau (\alpha)/M^ 2_{\alpha}]=2\sqrt{2\lambda}/sh 2\sqrt{2\lambda}, \] showing that it has the same law as the first hitting time of 2 by a 3-dimensional Bessel process starting from 0.
This paper explains how this identity comes from path decompositions of the trajectories of Brownian paths, in the spirit of previous work by the same author.

Reviewer: Ph.Biane

MSC:

60J65

Brownian motion

Keywords:

Brownian motion; Bessel process; path decompositions of the trajectories of Brownian paths

Citations:

Zbl 0635.00013