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Sousa, Rúben; Guerra, Manuel; Yakubovich, Semyon

Lévy processes with respect to the Whittaker convolution. (English) Zbl 1478.60142

Trans. Am. Math. Soc. 374, No. 4, 2383-2419 (2021).
Summary: It is natural to ask whether it is possible to construct Lévy-like processes where actions by random elements of a given semigroup play the role of increments. Such semigroups induce a convolution-like algebra structure in the space of finite measures. In this paper, we show that the Whittaker convolution operator, related with the Shiryaev process, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. We then introduce the class of Lévy processes with respect to the Whittaker convolution and study their basic properties. We obtain a martingale characterization of the Shiryaev process analogous to Lévy’s characterization of Brownian motion. Our results demonstrate that a nice theory of Lévy processes with respect to generalized convolutions can be developed for differential operators whose associated convolution does not satisfy the usual compactness assumption on the support of the convolution.

Cited in 12 Documents

MSC:

60G51 Processes with independent increments; Lévy processes
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
47D07 Markov semigroups and applications to diffusion processes
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)

Keywords:

Lévy process; index Whittaker transform; generalized convolution; Shiryaev process; martingale characterization; infinitely divisible distributions

Software:

DLMF
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References:

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