Probability measures, Lévy measures and analyticity in time

Details Probability measures, Lévy measures and analyticity in time Probability measures, Lévy measures and analyticity in time

2008-11-05

arxiv.org/abs/0811.0678v1

Bernoulli 2008, Vol. 14, No. 3, 764-790

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/07-BEJ6114 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statist

Scientific paper

10.3150/07-BEJ6114

We investigate the relation of the semigroup probability density of an infinite activity L\'{e}vy process to the corresponding L\'{e}vy density. For subordinators, we provide three methods to compute the former from the latter. The first method is based on approximating compound Poisson distributions, the second method uses convolution integrals of the upper tail integral of the L\'{e}vy measure and the third method uses the analytic continuation of the L\'{e}vy density to a complex cone and contour integration. As a by-product, we investigate the smoothness of the semigroup density in time. Several concrete examples illustrate the three methods and our results.

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