Affine Dunkl processes

TL;DR

This paper introduces affine Dunkl processes for the affine root system of type A_1, constructing them via a skew-product decomposition, and studies their stochastic properties and jump behavior.

Contribution

It extends classical Dunkl processes to the affine setting, providing a new construction and analyzing their stochastic and martingale properties.

Findings

Affine Dunkl process is a cadlag Markov process.

The process has a well-defined jump structure.

It admits a martingale decomposition similar to classical Dunkl processes.

Abstract

We introduce the analogue of Dunkl processes in the case of an affine root system of type $\widetilde{\text{A}}_1$. The construction of the affine Dunkl process is achieved by a skew-product decomposition by means of its radial part and a jump process on the affine Weyl group, where the radial part of the affine Dunkl process is defined as the unique solution of some stochastic differential equation. We prove that the affine Dunkl process is a c\`adl\`ag Markov process as well as a local martingale, study its jumps, and give a martingale decomposition, which are properties similar to those of the classical Dunkl process.