International Journal of Stochastic Analysis
Volume 2014 (2014), Article ID 520136, 49 pages
Research Article

From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval

1Université de Lyon, Institut Camille Jordan, CNRS UMR5208, France
2Institut National des Sciences Appliquées de Lyon Pôle de Mathématiques, Bâtiment Léonard de Vinci, 20 Avenue Albert Einstein, 69621 Villeurbanne Cedex, France

Received 20 May 2013; Accepted 14 August 2013; Published 26 March 2014

Academic Editor: M. Lopez-Herrero

Copyright © 2014 Aimé Lachal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let be a positive integer, a positive constant and be a sequence of independent identically distributed pseudorandom variables. We assume that the ’s take their values in the discrete set and that their common pseudodistribution is characterized by the (positive or negative) real numbers for any . Let us finally introduce the associated pseudorandom walk defined on by and for . In this paper, we exhibit some properties of . In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudorandom walk to the pseudo-Brownian motion driven by the high-order heat-type equation . We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudoprocess.