We revisit a result of Uchiyama (1980): given that a certain integral test is satisfied, the rate of the probability that Brownian motion remains below the moving boundary f is asymptotically the same as for the constant boundary. The integral test for f is also necessary in some sense.After Uchiyama's result, a number of different proofs appeared simplifying the original arguments, which strongly rely on some known identities for Brownian motion. In particular, Novikov (1996) gives an elementary proof in the case of an increasing boundary. Here, we provide an elementary, half-page proof for the case of a decreasing boundary. Further, we identify that the integral test is related to a repulsion effect of the three-dimensional Bessel process. Our proof gives some hope to be generalized to other processes such as FBM.
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.
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