Large deviations for functionals of some self-similar Gaussian processes

Abstract

We prove large deviation principles for ${\int }_0^t\gamma (X_s)ds$, where $X$ is a $d$-dimensional self-similar Gaussian process and $\gamma (x)$ takes the form of the Dirac delta function $\delta (x)$, $|x{|}^{-\beta }$ with $\beta \in (0,d)$, or ${\prod }_{i=1}^d|x_i{|}^{-{\beta }_i}$ with ${\beta }_i\in (0,1)$. In particular, large deviations are obtained for the functionals of $d$-dimensional fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. As an application, the critical exponential integrability of the functionals is discussed.