Generalization of Doob Decomposition Theorem and Risk Assessment in Incomplete Markets

TL;DR

This paper generalizes the Doob decomposition theorem for supermartingales relative to convex sets of measures, introduces the concept of complete measure sets, and proposes a new fair price definition for options in incomplete markets.

Contribution

It extends the Doob decomposition to supermartingales under convex measure sets and defines fair prices in incomplete markets.

Findings

Characterization of local regular supermartingales

Complete sets of measures ensure supermartingale regularity

Derived a formula for European option fair price

Abstract

In the paper, we introduce the notion of a local regular supermartingale relative to a convex set of equivalent measures and prove for it the necessary and sufficient conditions of optional Doob decomposition in the discrete case. This Theorem is a generalization of the famous Doob decomposition onto the case of supermartingales relative to a convex set of equivalent measures. The description of all local regular supermartingales relative to a convex set of equivalent measures is presented. A notion of complete set of equivalent measures is introduced. We prove that every non negative bounded supermartingale relative to a complete set of equivalent measures is local regular. A new definition of fair price of contingent claim in incomplete market is given and a formula for fair price of Standard option of European type is found.