We provide a general approach to obtain upper bounds for small deviations $ \mathbb{ P}(\Vert y \Vert \le \epsilon)$ in different norms, namely the supremum and $\beta$- H\"older norms. The large class of processes $y$ under consideration takes the form $y_t= X_t + \int_0^t a_s d s$, where $X$ and $a$ are two possibly dependent stochastic processes. Our approach provides an upper bound for small deviations whenever upper bounds for the \textit{ concentration of measures} of $L^p$- norm of random vectors built from increments of the process $X$ and \textit{ large deviation} estimates for the process $a$ are available. Using our method, among others, we obtain the optimal rates of small deviations in supremum and $\beta$- H\"older norms for fractional Brownian motion with Hurst parameter $H\le\ \frac{ 1}{ 2}$. As an application, we discuss the usefulness of our upper bounds for small deviations in pathwise stochastic integral representation of random variables motivated by the hedging problem in mathematical finance.
Categories: math.PR