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 Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 2, Pages 140–171 (Mi izv8575)

Integrals of Bessel processes and multi-dimensional Ornstein–Uhlenbeck processes: exact asymptotics for $L^p$-functionals

V. R. Fatalov

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove results on exact asymptotics of the expectations $\mathbf{E}_a \exp (-\int_0^T \xi_q^p(t) dt )$, $\mathbf{E}_a [ \exp (-\int_0^T \xi_q^p(t) dt ) | \xi_q(T)=b ]$ as $T\to\infty$ for $p>0$, $a\geq 0$, $b\geq 0$, where $\xi_q(t)$, $t\geq 0$, is a Bessel process of order $q\geq-1/2$. We also find exact asymptotics of the probabilities $\mathbf{P} \{ \int_0^1 \sum_{k=1}^n |Y_k(t)|^p dt \leq \varepsilon^p \}$, $\mathbf{P} \{ \int_0^1 [ \sum_{k=1}^n Y_k^2(t) ]^{p/2} dt \leq \varepsilon^p \}$ as $\varepsilon\to 0$, where $\mathbf{Y}(t)=(Y_1(t),…, Y_n(t))$, $t\geq 0$, is the $n$-dimensional non-stationary Ornstein–Uhlenbeck process with a parameter $\gamma=(\gamma_1, …, \gamma_n)$ starting at the origin. We also obtain a number of other results. Numerical values of the asymptotics are given for $p=1$, $p=2$.

Keywords: Bessel processes, Feynman–Kac formula, multi-dimensional Wiener process, Girsanov's theorem, small deviations, Schrödinger operator, Airy function, Bessel function.

 Funding Agency Grant Number Russian Foundation for Basic Research 11-01-00050 This paper was written with the support of the Russian Foundation for Basic Research (grant no. 11-01-00050).

DOI: https://doi.org/10.4213/im8575

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English version:
Izvestiya: Mathematics, 2018, 82:2, 377–406

Bibliographic databases:

Document Type: Article
UDC: 519.21
MSC: 60F25, 60J25
Revised: 12.08.2016

Citation: V. R. Fatalov, “Integrals of Bessel processes and multi-dimensional Ornstein–Uhlenbeck processes: exact asymptotics for $L^p$-functionals”, Izv. RAN. Ser. Mat., 82:2 (2018), 140–171; Izv. Math., 82:2 (2018), 377–406

Citation in format AMSBIB
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