Mathematics – Probability
Scientific paper
2011-12-19
Mathematics
Probability
20 pages, 10 figures
Scientific paper
We study the density of the supremum of a strictly stable L\'evy process. As was proved recently in F. Hubalek and A. Kuznetsov "A convergent series representation for the density of the supremum of a stable process" (Elect. Comm. in Probab., 16, 84-95, 2011), for almost all irrational values of the stability parameter $\alpha$ this density can be represented by an absolutely convergent series. We show that this result is not valid for all irrational values of $\alpha$: we construct a dense uncountable set of irrational numbers $\alpha$ for which the series does not converge absolutely. Our second goal is to investigate in more detail the important case when $\alpha$ is rational. We derive an explicit formula for the Mellin transform of the supremum, which is given in terms of Gamma function and dilogarithm. In order to illustrate the usefulness of these results we perform several numerical experiments and discuss their implications. Finally, we state some interesting connections that this problem has to other areas of Mathematics and Mathematical Physics, such as q-series, Diophantine approximations and quantum dilogarithms, and we also suggest several open problems.
No associations
LandOfFree
On the density of the supremum of a stable process does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the density of the supremum of a stable process, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the density of the supremum of a stable process will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-209789