An exact asymptotics for small deviations of a nonstationary Ornstein-Uhlenbeck process in the L p -norm, p ≥ 2

Abstract

A result concerning an exact asymptotics for the probability \(P\{ \int\limits_0^1 {|\zeta _\gamma (t)|^p dt \leqslant \varepsilon ^p } \} ,\varepsilon \to 0\), where p ≥ 2, is proved for a nonstationary Gaussian Markov process ζγ(t) of Ornstein-Uhlenbeck with zero mean and the covariance function \(E\zeta _\gamma (t)\zeta _\gamma (s) = \tfrac{1}{{2\gamma }}[e^{ - \gamma |t - s|} - e^{ - \gamma |t + s|} ]\), s, t ≥ 0, where γ > 0 is a parameter. Investigation techniques are the Laplace method for the sojourn time of continuous-time Markov processes and reduction to the case of Wiener processes.