First Exit Time from a Bounded Interval for a Certain Class of Additive Functionals of Brownian Motion

Aimé Lachal ()
Additional contact information
Aimé Lachal: Institut National des Sciences Appliquées de Lyon

Journal of Theoretical Probability, 2000, vol. 13, issue 3, 733-775

Abstract: Abstract Let (B t) t≥0 be standard Brownian motion starting at y, X t = x + ∫ t 0 V(B s) ds for x ∈ (a, b), with V(y) = y γ if y≥0, V(y)=−K(−y) γ if y≤0, where γ>0 and K is a given positive constant. Set τ ab=inf{t>0: X t∉(a, b)} and σ 0=inf{t>0: B t=0}. In this paper we give several informations about the random variable τ ab. We namely evaluate the moments of the random variables $$B_{\tau _{ab} } and B_{\tau _{ab} \wedge \sigma _0 } $$ , and also show how to calculate the expectations $${\mathbb{E}}\left( {\tau _{ab}^m B_{\tau _{ab} }^n } \right) and {\mathbb{E}}\left( {\left( {\tau _{ab} \wedge \sigma _0 } \right)^m B_{\tau _{ab} \wedge \sigma _0 }^n } \right)$$ . Then, we explicitly determine the probability laws of the random variables $$B_{{\tau }_{ab} } and B_{{\tau }_{ab} \wedge \sigma _0 }$$ as well as the probability $${\mathbb{P}}\left\{ {X_{\tau _{ab} } = a\left( {or b} \right)} \right\}$$ by means of special functions.

Keywords: first exit time; excursion process; Abel's integral equation; hypergeometric functions (search for similar items in EconPapers)
Date: 2000
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
http://link.springer.com/10.1023/A:1007810528683 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:13:y:2000:i:3:d:10.1023_a:1007810528683

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1023/A:1007810528683

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().