  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.
