L\'evy processes with respect to the index Whittaker convolution

TL;DR

This paper introduces a new class of Le9vy processes based on the index Whittaker convolution, exploring their properties and connecting them to the Shiryaev process, thus expanding the theory of generalized convolutions.

Contribution

It develops the theory of Le9vy processes with respect to the index Whittaker convolution, including a martingale characterization of the Shiryaev process.

Findings

The square root of the Shiryaev process is a Le9vy process in this framework.

The theory extends to convolutions lacking compact support, linking algebraic properties to differential operator singularities.

Provides a martingale characterization of the Shiryaev process similar to Brownian motion.

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.