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From Pseudorandom Walk to Pseudo-Brownian Motion : First Exit Time from a One-Sided or a Two-Sided Interval
Author
Source
International Journal of Stochastic Analysis
Issue
Vol. 2014, Issue 2014 (31 Dec. 2014), pp.1-49, 49 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2014-03-26
Country of Publication
Egypt
No. of Pages
49
Main Subjects
Abstract EN
Let N be a positive integer, c a positive constant and (ξn)n≥1 be a sequence of independent identically distributed pseudorandom variables.
We assume that the ξn’s take their values in the discrete set {-N,-N+1,…,N-1,N} and that their common pseudodistribution is characterized by the (positive or negative) real numbers ℙ{ξn=k}=δk0+(-1)k-1c(2Nk+N) for any k∈{-N,-N+1,…,N-1,N}.
Let us finally introduce (Sn)n≥0 the associated pseudorandom walk defined on ℤ by S0=0 and Sn=∑j=1nξj for n≥1.
In this paper, we exhibit some properties of (Sn)n≥0.
In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for (Sn)n≥0 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 ∂/∂t=(-1)N-1c∂2N/∂x2N.
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.
American Psychological Association (APA)
Lachal, Aimé. 2014. From Pseudorandom Walk to Pseudo-Brownian Motion : First Exit Time from a One-Sided or a Two-Sided Interval. International Journal of Stochastic Analysis،Vol. 2014, no. 2014, pp.1-49.
https://search.emarefa.net/detail/BIM-478224
Modern Language Association (MLA)
Lachal, Aimé. From Pseudorandom Walk to Pseudo-Brownian Motion : First Exit Time from a One-Sided or a Two-Sided Interval. International Journal of Stochastic Analysis No. 2014 (2014), pp.1-49.
https://search.emarefa.net/detail/BIM-478224
American Medical Association (AMA)
Lachal, Aimé. From Pseudorandom Walk to Pseudo-Brownian Motion : First Exit Time from a One-Sided or a Two-Sided Interval. International Journal of Stochastic Analysis. 2014. Vol. 2014, no. 2014, pp.1-49.
https://search.emarefa.net/detail/BIM-478224
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-478224