Small Deviations for Fractional Stable Processes Mikhail Lifshits and Thomas Simon
Let {Rt , 0 ≤ t ≤ 1} be a symmetric α-stable Riemann-Liouville process with Hurst parameter H > 0. Consider a translation invariant, β-self-similar, and p-pseudo-additive functional semi-norm ||.||. We show that if H > β + 1/p and γ = (H − β − 1/p)−1 , then lim rγ log P {||R|| ≤ r} = −K ∈ [−∞, 0), r↓0
with K finite in the Gaussian case α = 2. If α < 2, we prove that K is finite when R is continuous and H > β + 1/p + 1/α. We also show that under the above assumptions, lim rγ log P {||X|| ≤ r} = −K ∈ (−∞, 0), r↓0
where X is the linear α-stable fractional motion with Hurst parameter H ∈ (0, 1) (if α = 2, then X is the classical fractional Brownian motion). These general results cover many cases previously studied in the literature, and also prove the existence of new small deviation constants, both in Gaussian and Non-Gaussian frameworks.
Small Deviations for Fractional Stable Processes ...
Let {Rt, 0 ⤠t ⤠1} be a symmetric α-stable Riemann-Liouville process with Hurst parameter H > 0. Consider a translation invari- ant, β-self-similar, and ...
provided the weight function Ï satisfies a condition slightly stronger than the râintegrability. Thus we extend earlier results for Brown- ian motion, i.e. H = 1/2, to the fractional case. Our basic tools are entropy estimates for fractional integ
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May 3, 2016 - Theoretically, roughness relates to the degree of Hölder regularity ... example of a process that can exhibit various degrees of roughness.
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As an illustration, let us consider the trivial ... We first briefly recall some basic facts about stochastic calculus with respect to a frac- tional Brownian motion.
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Introduction. We study the .... As an illustration, let us consider the trivial ... We first briefly recall some basic facts about stochastic calculus with respect to a frac-.