Martingales and Super-martingales Relative to a Convex Set of Equivalent Measures

TL;DR

This paper systematically studies martingales and super-martingales relative to convex sets of equivalent measures, introducing local regularity, and generalizing Doob's decomposition, with applications to fair pricing in incomplete markets.

Contribution

It introduces the concept of local regular super-martingales relative to convex measure sets and generalizes Doob's decomposition for these cases.

Findings

Characterization of local regular super-martingales

Conditions for super-martingales to be local regular

Formula for fair price of European options in incomplete markets

Abstract

In the paper, the martingales and super-martingales relative to a convex set of equivalent measures are systematically studied. The notion of local regular super-martingale relative to a convex set of equivalent measures is introduced and the necessary and sufficient conditions of the local regularity of it in the discrete case are founded. The description of all local regular super-martingales relative to a convex set of equivalent measures is presented. The notion of the complete set of equivalent measures is introduced. We prove that every bounded in some sense super-martingale relative to the complete set of equivalent measures is local regular. A new definition of the fair price of contingent claim in an incomplete market is given and the formula for the fair price of Standard Option of European type is found. The proved Theorems are the generalization of the famous Doob decomposition for super-martingale onto the case of super-martingales relative to a convex set of equivalent measures.