Bridges of quadratic harnesses

TL;DR

This paper demonstrates that bridges of quadratic harnesses can be transformed into standard quadratic harnesses via affine transformations, and characterizes those arising from Levy processes and q-Meixner processes.

Contribution

It establishes a comprehensive affine transformation framework for quadratic harness bridges and characterizes their connections to Levy and q-Meixner processes.

Findings

Bridges of quadratic harnesses can be transformed into standard forms.

Characterization of quadratic harnesses from Levy process bridges.

Identification of quadratic harnesses from stitched q-Meixner processes.

Abstract

Quadratic harnesses are typically non-homogeneous Markov processes with time-dependent state space. Using an appropriately defined affine transformation we show that all bridges of a given quadratic harness can be transformed into other standard quadratic harnesses. Conversely, each such bridge is an affine transformation of a standard quadratic harness. We describe quadratic harnesses that correspond to bridges of some Levy processes. We determine all quadratic harnesses that may arise from stitching together a pair of q-Meixner processes.