Simple examples of pure-jump strict local martingales

TL;DR

This paper introduces simple pure-jump strict local martingales constructed from exponentials of self-exciting affine Markov processes, providing criteria for their strict local martingale property and alternative measure change constructions.

Contribution

It offers new explicit examples of pure-jump strict local martingales and characterizes their properties through integral criteria and differential equation non-uniqueness.

Findings

Characterization of strict local martingale property via integral criterion

Construction of examples using exponentials of affine Markov processes

Alternative measure change approach for the examples

Abstract

We present simple new examples of pure-jump strict local martingales. The examples are constructed as exponentials of self-exciting affine Markov processes. We characterize the strict local martingale property of these processes by an integral criterion and by non-uniqueness of an associated ordinary differential equation. Finally we show an alternative construction for our examples by an absolutely continuous measure change in the spirit of (Delbaen and Schachermayer, PTRF 1995).