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Exact asymptotics of small deviations for a stationary Ornstein-Uhlenbeck process and some Gaussian diffusion processes in the L p-norm, 2 ≤ p ≤ ∞

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Abstract

We prove results on exact asymptotics of the probabilities

$$ P\left\{ {\int\limits_0^1 {\left| {\eta (t)} \right|^p dt \leqslant \varepsilon ^p } } \right\},\varepsilon \to 0, $$

where 2 ≤ p ≤ ∞, for two types of Gaussian processes η(t), namely, a stationary Ornstein-Uhlenbeck process and a Gaussian diffusion process satisfying the stochastic differential equation

$$ \left\{ \begin{gathered} dZ(t) = dw(t) + g(t)Z(t)dt,t \in [0,1], \hfill \\ Z(0) = 0. \hfill \\ \end{gathered} \right. $$

Derivation of the results is based on the principle of comparison with a Wiener process and Girsanov’s absolute continuity theorem.

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  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    V. R. Fatalov

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Correspondence to V. R. Fatalov.

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Original Russian Text © V.R. Fatalov, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 2, pp. 75–95.

Supported in part by the Russian Foundation for Basic Research, project no. 04-01-00700.

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Fatalov, V.R. Exact asymptotics of small deviations for a stationary Ornstein-Uhlenbeck process and some Gaussian diffusion processes in the L p-norm, 2 ≤ p ≤ ∞. Probl Inf Transm 44, 138–155 (2008). https://doi.org/10.1134/S0032946008020063

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