On the existence of an equivalent supermartingale density for a fork-convex family of stochastic processes
type: Item
instance of: scholarly article
title: On the existence of an equivalent supermartingale density for a fork-convex family of stochastic processes
zbMATH Open document ID: 1203.60046
DOI: 10.1134/S0001434610030338
published in: Mathematical Notes
publication date: 2010-09-24T00:00:00Z
review text: The author introduces the notion of fork-convex family of stochastic processes that in some sense generalizes the usual notion of convexity. It is proved that a fork-convex family \(\mathbb{W}\) admits an equivalent supermartingale density if and only if the set \(H\) of nonnegative random variables majorized by the values of elements of \(\mathbb{W}\) at fixed instants of time is bounded in probability. This result has a clear interpretation in the framework of the mathematical theory of arbitrage, and what is more, the number of assets can be arbitrary. Namely, the notion of No Unbounded Profit with Bounded Risk (NUPBR) is considered and it is proved that the NUPBR condition is satisfied if and only if there exists an equivalent supermartingale density for the corresponding set of nonnegative stochastic integrals with respect to the semimartingales.
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60G48
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 91B24
zbMATH DE Number: 5791084
zbMATH Keywords: fork-convex family of nonnegative stochastic processes
zbMATH Keywords: equivalent supermartingale density
zbMATH Keywords: no unbounded profit with bounded risk
author: D. B. Rokhlin
reviewed by: Yuliya S. Mishura
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5791084