2003/10/26 by G. M. Molchan, G. Molchan, Molchan, G. +2 · 1 citation
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60G15 #60G18 #Complex Systems and Time Series Analysis #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:60G15 #msc:60G18
paper · pdf · doi:10.48550/arxiv.math/0310413
15 pages, 4 figures
arxiv created 2003/10/26 · openalex publication_date 2003/10/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the paths of a Gaussian random process x(t), x(0)=0 not exceeding a fixed positive level over a large time interval (0,T), T≫ 1. The probability p(T) of such event is frequently a regularly varying function at ∞ with exponent θ. In applications this parameter can provide information on fractal properties of processes that are subordinate to x(⋅). For this reason the estimation of θ is an important theoretical problem. Here, we consider the process x(t) whose derivative is fractional Brownian motion with self-similarity parameter 0<H<1. For this case we produce new computational evidence in favor of the relations log p(T)=-θlog T(1+o(1)) and θ=H(1-H). The estimates of θ are to within 0.01 in the range 0.1≤ H≤ 0.9. An analytical result for the problem in hand is known for the markovian case alone, i.e., for H=1/2. We point out other statistics of x(t) whose small values have probabilities of the same order as p(T) in the log scale.