TITLE:
Martingales and Super-Martingales Relative to a Convex Set of Equivalent Measures
AUTHORS:
Nicholas S. Gonchar
KEYWORDS:
Random Process, Convex Set of Equivalent Measures, Optional Doob Decomposition, Local Regular Super-Martingale, Martingale, Fair Price of Contingent Claim
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.8 No.4,
April
24,
2018
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.