Stitching pairs of Levy processes into harnesses

TL;DR

This paper explores how Levy processes within exponential families can be combined into a single harness through deterministic reparametrization and boundary law selection, with special cases forming quadratic harnesses.

Contribution

It introduces a method to stitch Levy processes into harnesses, including quadratic harnesses, using reparametrization and boundary law choices.

Findings

Pairs of Levy processes can be stitched into harnesses.

Stitching preserves Markov property under certain conditions.

Quadratic harnesses can be constructed from Levy-Meixner processes.

Abstract

We consider natural exponential families of Levy processes with randomized parameter. Such processes are Markov, and under suitable assumptions, pairs of such processes with shared randomization can be stitched together into a single harness. The stitching consists of deterministic reparametrization of the time for both processes, so that they run on adjacent time intervals, and of the choice of the appropriate law at the boundary. Processes in the Levy-Meixner class have an additional property that they are quadratic harnesses, and in this case stitching constructions produce quadratic harnesses.