arXiv subjects

Tanja Kramm

Publications and source records attributed to Tanja Kramm.

3 recordsLinked to original sources

arXiv2015

The first passage time problem over a moving boundary for asymptotically stable Lévy processes

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 $- α$.

math.PR↗

arXiv2012

First passage times of Lévy processes over a one-sided moving boundary

We study the asymptotic behaviour of the tail of the distribution of the first passage time of a Lévy process over a one-sided moving boundary. Our main result states that if the boundary behaves as $t^γ$ for large $t$ for some $γ<1/2$ then the probability that the process stays below the boundary behaves asymptotically as in the case of a constant boundary. We do not have to assume Spitzer's condition in contrast to all previously known results. Both positive ($+t^γ$) and negative ($-t^γ$) boundaries are considered. These results extend the findings of Greenwood and Novikov (1986) and are also motivated by results in the case of Brownian motion, for which the above result was proved in Uchiyama (1980).

math.PR↗

arXiv2012

First exit of Brownian motion from a one-sided moving boundary

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.

math.PR↗