Strict local martingales: examples

TL;DR

This paper characterizes strict local martingales through the integrability of their supremum process and constructs a family of such martingales based on this criterion.

Contribution

It provides a new criterion for identifying strict local martingales and offers explicit examples demonstrating this property.

Findings

Strict local martingales have supremum processes not in $L_eta$ for some $eta<1$

Constructs a family of strict local martingales based on the supremum process criterion

Provides a theoretical framework linking supremum process integrability to martingale properties

Abstract

We show that a continuous local martingale is a strict local martingale if its supremum process is not in $ L_\alpha$ for a positive number $\alpha $ smaller than $1$. Using this we construct a family of strict local martingales.