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L\'evy processes with respect to the index Whittaker convolution
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In this article, the authors introduced the class of L\'evy processes with respect to the index Whittaker convolution and studied their basic properties, and showed that the square root of the Shiryaev process belongs to the family of L'evy process.Abstract:
The index Whittaker convolution operator, recently introduced by the authors, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. In this paper, we introduce the class of L\'evy processes with respect to the index Whittaker convolution and study their basic properties. We prove that the square root of the Shiryaev process belongs to our family of L\'evy process, and this is shown to yield a martingale characterization of the Shiryaev process analogous to L\'evy's characterization of Brownian motion.
Our results demonstrate that a nice theory of L\'evy processes with respect to generalized convolutions can be developed even if the usual compactness assumption on the support of the convolution fails, shedding light into the connection between the properties of the convolution algebra and the nature of the singularities of the associated differential operator.read more
Citations
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References
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Lévy processes and infinitely divisible distributions
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Journal ArticleDOI
The special functions and their approximations
C. W. Clenshaw,Yudell L. Luke +1 more