A Few Remarks on the Supremum of Stable Processes
نویسنده
چکیده
In [1], Bernyk et al. offer a power series and an integral representation for the density of S1, the maximum up to time 1, of a regular spectrally positive α-stable Lévy process. They also state the asymptotic behavior for large values of the density. A fact which was proved by Doney [4], by investigating the integral representation. In this note, we provide the asymptotic expansion of the density and of its successive derivatives from the power series representation. We also show that the density of the positive random variable S−α 1 is the Laplace transform of a function which takes negative values on R and thus it is not completely monotone.
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