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

Exact asymptotics of small deviations for a stationary Ornstein-Uhlenbeck process and some... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

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

, Volume 44 (2) – Jul 11, 2008
18 pages

/lp/springer_journal/exact-asymptotics-of-small-deviations-for-a-stationary-ornstein-Dr81LlXsDg
Publisher
Springer Journals
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946008020063
Publisher site
See Article on Publisher Site

### 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.

### Journal

Problems of Information TransmissionSpringer Journals

Published: Jul 11, 2008

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