Analysis of continuous strict local martingales via h-transforms

TL;DR

This paper investigates strict local martingales using h-transforms, identifying conditions for their existence, providing examples, and exploring implications for financial bubbles and non-standard behaviors.

Contribution

It introduces a new perspective on strict local martingales via h-transforms, including multidimensional cases and diverse examples, expanding understanding of their properties and behaviors.

Findings

Strict local martingales arise from non-equivalent measure changes.

Examples include diffusions with various boundary behaviors and non-colliding diffusions.

Strict local martingales exhibit non-uniform behavior under certain functions.

Abstract

We study strict local martingales via h-transforms, a method which first appeared in Delbaen-Schachermayer. We show that strict local martingales arise whenever there is a consistent family of change of measures where the two measures are not equivalent to one another. Several old and new strict local martingales are identified. We treat examples of diffusions with various boundary behavior, size-bias sampling of diffusion paths, and non-colliding diffusions. A multidimensional generalization to conformal strict local martingales is achieved through Kelvin transform. As curious examples of non-standard behavior, we show by various examples that strict local martingales do not behave uniformly when the function (x-K)^+ is applied to them. Implications to the recent literature on financial bubbles are discussed.