Doob's decomposition of set-valued submartingales via ordered near vector spaces

Abstract

We use ideas from measure-free martingale theory and Rådström's completion of a near vector space to derive a Doob decomposition of submartingales in ordered near vector spaces. As a special cases thereof, we obtain the Doob decomposition of set-valued submartingales, as noted by Daures, Ni and Zhang, and an analogue of the Doob decomposition of set-valued submartingales, as noted by Shen and Wang.