zbMATH — the first resource for mathematics

A diffusion process stopped at the first time its amplitude reaches a fixed level. (Diffusion arrêtée au premier instant où l’amplitude atteint un niveau donné.) (French) Zbl 0808.60069
The amplitude of a real process \(x_ t\) is defined to be the difference between its maximum and minimum process. For the cases of \(x_ t\) being a Brownian motion, a continuous diffusion or the angular part of a planar Brownian motion, the distribution of the first time when the amplitude reaches a pregiven level is studied and characterized.
Reviewer: M.Kohlmann (Bonn)

60J60 Diffusion processes
Full Text: DOI
[1] Feller W., The asymptotic distribution or the range of sums of independent random variables 22 pp 427– (1951) · Zbl 0043.34201
[2] Feller W., An introduction to probability theory and its applications · Zbl 0039.13201
[3] Imhof J. P., On the range of Brownian motion and its inverse process 13 (3) pp 1011– (1985) · Zbl 0579.60084
[4] Imhof J. P., A construction of the Brownian path from BES(3) pieces 13 (3) (1992) · Zbl 0762.60071
[5] Le Gall J. F., Points cônes du mouvement brownien plan le cas critique 93 (3) pp 231– (1992) · Zbl 0767.60084
[6] Pitman J., Asymptotic laws of planar Brownian motion 14 (3) pp 733– (1986) · Zbl 0607.60070
[7] Pitman J., Level crossings of a Cauchy process 14 (3) pp 780– (1986) · Zbl 0602.60059
[8] Spitzer F., Some theorems on 2-dimensional Brownian motion 87 (3) pp 187– (1958) · Zbl 0089.13601
[9] Vallois P., Une extension des théorèmes de Ray et Knight sur les temps locaux browniens 88 (3) pp 445– (1991) · Zbl 0723.60097
[10] Williams D., Decomposing the Brownian path 76 (3) pp 871– (1970) · Zbl 0233.60066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.