A remark on H1 martingales
Hardy Hulley, Johannes Ruf
TL;DR
This paper explores the relationship between H1 martingales and uniformly integrable martingales, providing a systematic way to construct examples of uniformly integrable martingales that are not H1 martingales.
Contribution
It offers a systematic template for constructing uniformly integrable martingales that are not H1 martingales, clarifying their distinction.
Findings
H1 martingales are a subset of uniformly integrable martingales.
Not all uniformly integrable martingales are H1 martingales.
A systematic construction method for non-H1 uniformly integrable martingales.
Abstract
The space of H1 martingales is interesting because of its duality with the space of BMO martingales. It is straightforward to show that every H1 martingale is a uniformly integrable martingale. However, the converse is not true. That is to say, some uniformly integrable martingales are not H1 martingales. This brief note provides a template for systematically constructing such processes.
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Taxonomy
TopicsAdvanced Harmonic Analysis Research · Advanced Banach Space Theory · Stochastic processes and financial applications