Martingale Property in Terms of Semimartingale Problems

TL;DR

This paper investigates when non-negative local martingales are true martingales using semimartingale theory, providing new characterizations, existence, and uniqueness results, including conditions for stochastic exponentials driven by infinite-dimensional Brownian motion.

Contribution

It offers a semimartingale perspective on the martingale property, connecting it to semimartingale problem solutions and deriving explicit conditions in infinite-dimensional settings.

Findings

Characterization of martingale property via semimartingale problems

New existence and uniqueness results for semimartingale problems

Explicit conditions for stochastic exponentials driven by infinite-dimensional Brownian motion

Abstract

Starting from the seventies mathematicians face the question whether a non-negative local martingale is a true or a strict local martingale. In this article we answer this question from a semimartingale perspective. We connect the martingale property to existence, uniqueness and topological properties of semimartingale problems. This not only leads to valuable characterizations of the martingale property, but also reveals new existence and uniqueness results for semimartingale problems. As a case study we derive explicit conditions for the martingale property of stochastic exponentials driven by infinite-dimensional Brownian motion.