2014/07/14 by Ehsan Azmoodeh, Lauri Viitasaari, Azmoodeh, Ehsan +1
Economics, Econometrics and Finance · Mathematics · #60F10 #60G15 #60G22 #60G50 #91G99 #Complex Systems and Time Series Analysis #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:60F10 #msc:60G15 #msc:60G22 #msc:60G50 #msc:91G99
paper · pdf · doi:10.48550/arxiv.1407.3553
openalex publication_date 2014/07/14 · arxiv created 2015/02/17 · arxiv updated 2015/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a general approach to obtain upper bounds for small deviations ℙ(\Vert y \Vert ≤ ε) in different norms, namely the supremum and β- Hölder norms. The large class of processes y under consideration takes the form yt= Xt + ∫0t as 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 concentration of measures of Lp- norm of random vectors built from increments of the process X and 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 β- Hölder norms for fractional Brownian motion with Hurst parameter H≤ (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.