2018/05/08 by Rúben Sousa, Sousa, Rúben, Manuel Guerra +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #33C15 #47D07 #60B15 #60G51 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1805.03051
openalex publication_date 2018/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Lévy 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 Lévy process, and this is shown to yield 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 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.