The joint Laplace transform of $T$ and $X(T)$ is derived where $X(\bullet)$ is a time homogeneous diffusion process and $T$ is the first time the process falls a specified amount below its current maximum. This generalizes the work of Taylor. The distribution of the maximum at $T$ is shown to be exponential for Brownian motion. Formulas for more general stopping times based on the current maximum are given.
Publié le : 1977-08-14
Classification:
Diffusion process,
stopping time,
Laplace transform,
stochastic differential equation,
60J60,
60G40,
60H10
@article{1176995770,
author = {Lehoczky, John P.},
title = {Formulas for Stopped Diffusion Processes with Stopping Times Based on the Maximum},
journal = {Ann. Probab.},
volume = {5},
number = {6},
year = {1977},
pages = { 601-607},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995770}
}
Lehoczky, John P. Formulas for Stopped Diffusion Processes with Stopping Times Based on the Maximum. Ann. Probab., Tome 5 (1977) no. 6, pp. 601-607. http://gdmltest.u-ga.fr/item/1176995770/