Small deviations in p-variation for stable processes

الأصل · EN

Let {Zₜ, t≥ 0} be a strictly stable process on with index α∈ (0,2]. We prove that for every p > α, there exists γ= γ(α, p) and = (α, p)∈ (0, +∞) such that 0γ ||Z||ₚ≤ = -, where ||Z||ₚ stands for the strong p-variation of Z on [0,1]. The critical exponent γ(α, p) takes a different shape according as |Z| is a subordinator and p >1, or not. The small ball constant (α, p) is explicitly computed when p ≤ 1, and a lower bound on (α, p) is easily obtained in the general case. In the symmetric case and when p > 2, we can also give an upper bound on (α, p) in terms of the Brownian small ball constant under the (1/p)-Hölder semi-norm. Along the way, we remark that the positive random variable ||Z||ᵖₚ is not necessarily stable when p > 1, which gives a negative answer to an old question of P. E. Greenwood.