Dominated families of martingale, supermartingale and quasimartingale laws
type: Item
instance of: scholarly article
title: Dominated families of martingale, supermartingale and quasimartingale laws
zbMATH Open document ID: 0908.60027
DOI: 10.1016/0304-4149(96)00073-7
published in: Stochastic Processes and their Applications
publication date: 1998-11-23T00:00:00Z
review text: The paper deals with two types of tamed (i.e. absolutely continuous with respect to its selected element) sets of probabilities on a probability space \((\Omega ,\mathcal F,P)\). The first type is formed by sets of probabilities \(Q << P\) such that a given convex set \(K\) of \(P\)-a.s. finite random variables (space \(L^{0}(\Omega ,\mathcal F,P)\)) has a uniformly finite \(Q\)-expectation. Such a set of probabilities contains an element equivalent to \(P\) if and only if a certain spanning condition (C) in the space \(L^{0}(\Omega ,\mathcal F,P)\) is satisfied. The second type contains probability sets which consist of quasi-, semi-, {super-,} sub- or simply martingale laws for an at most countable family \(\chi \) of adapted stochastic processes, i.e. sets of probabilities under which all members of \(\chi \) are quasi-, \dots , -martingales. It is shown that the second type set is also of the first type, provided that \(K\) is generated by stochastic integrals over \(\chi \), the integrands being elementary predictable processes with an additional property. Thus, a characterization of quasi-, \dots , -martingales is obtained, with an application to securities markets with frictions. An equivalent quasi-, \dots , -martingale measure exists on such a market if and only if there are no extreme arbitrage opportunities, a condition equivalent to (C).
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60G35
zbMATH DE Number: 1226253
zbMATH Keywords: semimartingale
zbMATH Keywords: quasimartingale law
zbMATH Keywords: equivalent probability measure
author: Marco Frittelli
reviewed by: Alexis Derviz
MaRDI profile type: Publication
full work available at URL: 0304-4149(96)00073-7
OpenAlex ID: W1994222882
cites work: Equivalent martingale measures and no-arbitrage in stochastic securities market models
cites work: REPRESENTING MARTINGALE MEASURES WHEN ASSET PRICES ARE CONTINUOUS AND BOUNDED
cites work: A general version of the fundamental theorem of asset pricing
cites work: Q3957682
cites work: Multiperiod security markets with differential information
cites work: Almost sure characterization of Martingales
cites work: Q5183990
cites work: Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics
cites work: Martingales and stochastic integrals in the theory of continuous trading
cites work: Calcul stochastique et problèmes de martingales
cites work: ARBITRAGE IN SECURITIES MARKETS WITH SHORT-SALES CONSTRAINTS
cites work: Arbitrage and equilibrium in economies with infinitely many commodities
cites work: Martingale Measures For A Class of Right‐Continuous Processes
cites work: MARTINGALE MEASURES FOR DISCRETE‐TIME PROCESSES WITH INFINITE HORIZON
cites work: Arbitrage et lois de martingale. (Arbitrage and martingale laws)
description: scientific article; zbMATH DE number 1226253