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
Springer Journals
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

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