The First Passage Time Problem Over a Moving Boundary for Asymptotically Stable Lévy Processes

Abstract

We study the asymptotic tail behaviour of the first-passage time over a moving boundary for asymptotically α-stable Lévy processes with α < 1.Our main result states that if the left tail of the Lévy measure is regularly varying with index −α and the moving boundary is equal to 1 − t γ for some γ < 1/α, then the probability that the process stays below the moving boundary has the same asymptotic polynomial order as in the case of a constant boundary. The same is true for the increasing boundary 1 + t γ with γ < 1/α under the assumption of a regularly varying right tail with index −α.

First-Passage Times over Moving Boundaries for Asymptotically Stable Walks

“…We note also that (11) is fulfilled if, for example, g n = O(c n / log 1+a n) with some a > 1/α(1 − ρ). A logarithmic version of this result has been shown by Aurzada and Kramm [2]. More precisely, they have proven that P(T g > n) = n ρ−1+o (1) for any boundary satisfying g n = O(n γ ) with some γ < 1/α.…”

Cramér’s estimate for stable processes with power drift

“…In this respect, we should mention that the condition γα > 1 on the drift power is optimal: in the Cauchy case α = γ = 1, the same Theorem A in [12] shows that the lower tail probability exponent depends on µ. Our argument relies in an essential way on the strong Markov property of the bidimensional process {(L (1) t , L t ), t ≥ 0} and is hence specific to the case β = 1. The other cases are believed to be challenging.…”

“…Recall e.g. from Lemma 14.11 and Theorem 14.19 in [15] that ρ ∈ [1 − 1/α, 1/α] for α ∈ [1,2] and ρ ∈ [0, 1] for α ∈ (0, 1), and that with this normalization, for α ∈ (0, 2) the Lévy measure of L has density ν(x) = Γ(1 + α) π sin(πα(1 − ρ)) |x| 1+α 1 {x<0} + sin(παρ) x 1+α 1 {x>0} . Throughout, we assume that L takes positive values i.e.…”

“…The lower bound is more involved and we will need the strong Markovian character of the twodimensional process {(L (1) t , L t ), t ≥ 0}, setting by P (x,y) for its law starting from (x, y) ∈ R 2 . Define the stopping time…”

An Exact Asymptotics for the Moment of Crossing a Curved Boundary by an Asymptotically Stable Random Walk

“…However, recent results by Aurzada and Kramm [2] give strong grounds to suspect that conditions in [9] are not optimal. More precisely, it is shown in [2] that if g(t) = o(t γ ) with some γ < 1/α, then P(T g > n) and P(T 0 > n) have the same rough asymptotics, that is,…”