Remarks on martingale representation theorem for set-valued martingales

TL;DR

This paper critically examines the martingale representation theorem for set-valued martingales, showing it only applies in trivial cases, and proposes a revised theorem for certain non-degenerate cases.

Contribution

It demonstrates the limitations of the existing theorem and introduces a revised version applicable to specific non-degenerate set-valued martingales.

Findings

Original theorem only applies to degenerate (point-valued) cases.

Revised theorem extends to certain non-degenerate set-valued martingales.

Highlights limitations in the current understanding of set-valued martingale representations.

Abstract

Martingale representation theorem for set-valued martingales was proposed by M. Kisielewicz [J. Math. Anal. Appl. 2014]. We shall prove that the result holds only for very special case: the set-valued martingale degenerates to the point-valued one. A revised representation theorem for a special kind of non-degenerate set-valued martingales is presented.