Optional Decomposition for continuous semimartingales under arbitrary filtrations

TL;DR

This paper provides an elementary approach to the Optional Decomposition Theorem for continuous semimartingales under general filtrations, requiring only the existence of strictly positive local martingale deflators rather than equivalent local martingale measures.

Contribution

It introduces a simplified proof of the Optional Decomposition Theorem applicable to broader filtrations without assuming equivalent local martingale measures.

Findings

Elementary proof of the Optional Decomposition Theorem

Applicable to general filtrations without equivalent measures

Relies on strictly positive local martingale deflators

Abstract

We present an elementary treatment of the Optional Decomposition Theorem for continuous semimartingales and general filtrations. This treatment does not assume the existence of equivalent local martingale measure(s), only that of strictly positive local martingale deflator(s).