A general approach to small deviation via concentration of measures

TL;DR

This paper introduces a unified method to bound small deviation probabilities for stochastic processes in various norms, leveraging concentration of measure and large deviation principles, with applications to fractional Brownian motion and finance.

Contribution

It develops a general framework for small deviation bounds applicable to a broad class of processes, including fractional Brownian motion, using concentration and large deviation techniques.

Findings

Derived optimal small deviation rates for fractional Brownian motion with Hurst parameter ≤ 1/2.

Provided bounds useful for stochastic integral representations in financial hedging.

Established a versatile approach applicable to dependent stochastic processes.

Abstract

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.