Generalization of Doob Decomposition Theorem and Risk Assessment in Incomplete Markets
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.
Keywords
Cite
@article{arxiv.1611.09062,
title = {Generalization of Doob Decomposition Theorem and Risk Assessment in Incomplete Markets},
author = {N. S. Gonchar},
journal= {arXiv preprint arXiv:1611.09062},
year = {2016}
}
Comments
arXiv admin note: substantial text overlap with arXiv:1601.03574