Single Jump Processes and Strict Local Martingales

TL;DR

This paper characterizes local martingales with a single jump at a random time, providing a framework to construct strict local martingales and related examples in stochastic analysis and finance.

Contribution

It introduces a classification of single jump local martingales based on deterministic functions and parameters, enabling new constructions of strict local martingales.

Findings

Characterization of local martingales with a single jump.

Construction methods for strict local martingales.

Examples of martingales that are not in H^1.

Abstract

Many results in stochastic analysis and mathematical finance involve local martingales. However, specific examples of strict local martingales are rare and analytically often rather unhandy. We study local martingales that follow a given deterministic function up to a random time $\gamma$ at which they jump and stay constant afterwards. The (local) martingale properties of these single jump local martingales are characterised in terms of conditions on the input parameters. This classification allows an easy construction of strict local martingales, uniformly integrable martingales that are not in $H^1$, etc. As an application, we provide a construction of a (uniformly integrable) martingale $M$ and a bounded (deterministic) integrand $H$ such that the stochastic integral $H\bullet M$ is a strict local martingale.