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 Probl. Peredachi Inf., 2007, Volume 43, Issue 3, Pages 75–96 (Mi ppi20)

Large Systems

Exact Asymptotics of Distributions of Integral Functionals of the Geometric Brownian Motion and Other Related Formulas

V. R. Fatalov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove results on exact asymptotics of the probabilities
$$\mathbf P\{\int\limits_0^1e^{\varepsilon\xi(t)} dt>b\},\qquad \mathbf P\{\int\limits_0^1e^{|\varepsilon\xi(t)|} dt>b\},\qquad \varepsilon\to0,$$
where $b>1$, for two Gaussian processes $\xi(t)$, 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.

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English version:
Problems of Information Transmission, 2007, 43:3, 233–254

Bibliographic databases:

UDC: 621.391.1:519.2

Citation: V. R. Fatalov, “Exact Asymptotics of Distributions of Integral Functionals of the Geometric Brownian Motion and Other Related Formulas”, Probl. Peredachi Inf., 43:3 (2007), 75–96; Problems Inform. Transmission, 43:3 (2007), 233–254

Citation in format AMSBIB
\Bibitem{Fat07} \by V.~R.~Fatalov \paper Exact Asymptotics of Distributions of Integral Functionals of the Geometric Brownian Motion and Other Related Formulas \jour Probl. Peredachi Inf. \yr 2007 \vol 43 \issue 3 \pages 75--96 \mathnet{http://mi.mathnet.ru/ppi20} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2360019} \zmath{https://zbmath.org/?q=an:1136.60357} \elib{http://elibrary.ru/item.asp?id=13544159} \transl \jour Problems Inform. Transmission \yr 2007 \vol 43 \issue 3 \pages 233--254 \crossref{https://doi.org/10.1134/S0032946007030064} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000255782800006} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-35848957680} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. R. Fatalov, “Exact asymptotics of Laplace-type Wiener integrals for $L^p$-functionals”, Izv. Math., 74:1 (2010), 189–216
2. V. R. Fatalov, “Integral Functionals for the Exponential of the Wiener Process and the Brownian Bridge: Exact Asymptotics and Legendre Functions”, Math. Notes, 92:1 (2012), 79–98
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