Strict Local Martingales with Jumps

TL;DR

This paper constructs explicit examples of strict local martingales with jumps, illustrating their emergence through filtration shrinkage, which is relevant in applications like finance, filtering, and control.

Contribution

It provides a novel method for constructing strict local martingales with jumps and analyzes their formation via filtration shrinkage.

Findings

Explicit examples of strict local martingales with jumps are provided.

Filtration shrinkage can generate strict local martingales in practical settings.

The paper offers a detailed analysis of the construction method.

Abstract

A strict local martingale is a local martingale which is not a martingale. There are few explicit examples of "naturally occurring" strict local martingales with jumps available in the literature. The purpose of this paper is to provide such examples, and to illustrate how they might arise via filtration shrinkage, a phenomenon we would contend is common in applications such as filtering, control, and especially in mathematical finance. We give a method for constructing such examples and analyze one particular method in detail.