The asymptotic behavior of densities related to the supremum of a stable process
Doney, R. A. ; Savov, M. S.
Ann. Probab., Tome 38 (2010) no. 1, p. 316-326 / Harvested from Project Euclid
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If X is a stable process of index α∈(0, 2) whose Lévy measure has density cx−α−1 on (0, ∞), and S1=sup0Xt, it is known that P(S1>x)∽Aα−1x−α as x→∞ and P(S1≤x)∽Bα−1ρ−1xαρ as x↓0. [Here ρ=P(X1>0) and A and B are known constants.] It is also known that S1 has a continuous density, m say. The main point of this note is to show that m(x)∽Ax−(α+1) as x→∞ and m(x)∽Bxαρ−1 as x↓0. Similar results are obtained for related densities.
Publié le : 2010-01-15
Classification:  Stable process,  stable meander,  supremum,  passage time density,  asymptotic behavior,  60J30,  60F15 
@article{1264434000,
     author = {Doney, R. A. and Savov, M. S.},
     title = {The asymptotic behavior of densities related to the supremum of a stable process},
     journal = {Ann. Probab.},
     volume = {38},
     number = {1},
     year = {2010},
     pages = { 316-326},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1264434000}
}

Doney, R. A.; Savov, M. S. The asymptotic behavior of densities related to the supremum of a stable process. Ann. Probab., Tome 38 (2010) no. 1, pp.  316-326. http://gdmltest.u-ga.fr/item/1264434000/