# Exact asymptotics of distributions of integral functionals of the geometric Brownian motion and other related formulas

Exact asymptotics of distributions of integral functionals of the geometric Brownian motion and... We prove results on exact asymptotics of the probabilities $$P\left\{ {\int\limits_0^1 {e^{\varepsilon \xi (t)} dt > b} } \right\},P\left\{ {\int\limits_0^1 {e^{\varepsilon |\xi (t)|} dt > b} } \right\},\varepsilon \to 0,$$ where b > 1, for two Gaussian processes ξ(t), namely, a Wiener process and a Brownian bridge. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm-Liouville type with the use of Legendre functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

# Exact asymptotics of distributions of integral functionals of the geometric Brownian motion and other related formulas

, Volume 43 (3) – Oct 26, 2007
22 pages

/lp/springer_journal/exact-asymptotics-of-distributions-of-integral-functionals-of-the-RXwlw3SlyY
Publisher
Nauka/Interperiodica
Subject
Engineering; Communications Engineering, Networks; Electronic and Computer Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946007030064
Publisher site
See Article on Publisher Site

### Abstract

We prove results on exact asymptotics of the probabilities $$P\left\{ {\int\limits_0^1 {e^{\varepsilon \xi (t)} dt > b} } \right\},P\left\{ {\int\limits_0^1 {e^{\varepsilon |\xi (t)|} dt > b} } \right\},\varepsilon \to 0,$$ where b > 1, for two Gaussian processes ξ(t), namely, a Wiener process and a Brownian bridge. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm-Liouville type with the use of Legendre functions.

### Journal

Problems of Information TransmissionSpringer Journals

Published: Oct 26, 2007

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations