A general approach to small deviation via concentration of measures

We provide a general approach to obtain upper bounds for small deviations $\mathbb{P}(\mathrm{\parallel }y\mathrm{\parallel }\le ϵ)$ in different norms, namely the supremum and $\beta$- H"older norms. The large class of processes $y$ under consideration takes the form $y_t=X_t+{\int }_0^t{a}_{s}ds$, 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 $\mathit{\text{large}}\mathit{\text{deviation}}$ 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.