The cumulant representation of the Lundberg root in the case of semicontinuous processes
Author:D. V. Gusak Translated by:Oleg Klesov Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 82 (2010).
Theor. Probability and Math. Statist. 82 (2011), 1-10
Primary 60G50; Secondary 60K10
August 2, 2011
MathSciNet review:2790479 Full-text PDF
Abstract: For the case of homogeneous processes , , , with independent increments and negative jumps, it is proved in A. V. Skorokhod, Random Processes with Independent Increments, Nauka, Moscow, 1964 that the functional
is a nondecreasing process with independent increments with respect to , and its moment generating function is expressed via the cumulant that satisfies the corresponding Lundberg equation. The corresponding representations of this cumulant are specified and its Lévy characteristics (namely, and Lévy's integral measure ) are evaluated by using some of the results of the author's work of 2007 for the processes under consideration.
V. Skorohod, Random processes with independent increments,
Mathematics and its Applications (Soviet Series), vol. 47, Kluwer
Academic Publishers Group, Dordrecht, 1991. Translated from the second
Russian edition by P. V. Malyshev. MR 1155400
D. V. Gusak, Boundary Problems for Processes with Independent Increments in Risk Theory, Trans. Institute of Mathematics, National Academy of Sciences of Ukraine, vol. 67, Kiev, 2007.
Lévy, Processus stochastiques et mouvement brownien,
Suivi d’une note de M. Loève. Deuxième édition
revue et augmentée, Gauthier-Villars & Cie, Paris, 1965
0190953 (32 #8363)