Profitability in a multiple strategy market
type: Item
instance of: scholarly article
title: Profitability in a multiple strategy market
zbMATH Open document ID: 1039.60045
author: Giacomo Aletti
author: Vincenzo Capasso
published in: Decisions in Economics and Finance
publication date: 2004-01-08T00:00:00Z
review text: Let \(({\mathcal F}_n)\) and \(({\mathcal G}_n)\) \((n\geq 0)\) be two filtrations on a probability space \((\Omega,{\mathcal F},P)\) such that \({\mathcal F}_n\subset{\mathcal G}_n\) \((n\geq 0)\). A sequence \((X_n)\) \((n\geq 0)\) is called an \(({\mathcal F}_n,{\mathcal G}_n)\)-martingale provided \((X_n)\) is integrable and \(({\mathcal F}_n)\)-adapted, and \(E[X_{n+1}- X_n\mid{\mathcal G}_n]= 0\) \((n\geq 0)\). A \(({\mathcal G}_n)\)-strategy is (by definition) a totally bounded \(({\mathcal G}_n)\)-previsible process \(C= (C_n)\) such that, for some integer \(N\geq 0\), \(C_n= 0\) a.s. for all \(n> N\). Let \(X= (X_n)\) be an integrable \(({\mathcal F}_n)\)-adapted process. A \(({\mathcal G}_n)\)-strategy \(C= (C_n)\) is called risk-free (for \(X\)) provided \[ \sum^\infty_{n=1} C_n(X_n- X_{n-1})\geq 0\;\text{a.s.}\quad\text{and}\quad P\Biggl( \sum^\infty_{n=1} C_n(X_n- X_{n-1})> 0\Biggr)> 0. \] Then the following (easily proved) extension of the (first) fundamental theorem of asset pricing holds: Let \(X= (X_n)\) be an \(({\mathcal F}_n)\)-adapted process. There exists a risk-free \(({\mathcal G}_n)\)-strategy for \(X\) iff there does not exist a probability measure \(\widetilde P\) equivalent to \(P\) such that \(X\) is an \(({\mathcal F}_n,{\mathcal G}_n)\)-martingale under \(\widetilde P\).
Mathematics Subject Classification ID: 60G48
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 91B28
zbMATH DE Number: 2021365
reviewed by: Klaus Schürger
MaRDI profile type: Publication
full work available at URL: s10203-003-0039-5
OpenAlex ID: W2082524136
cites work: Equivalent martingale measures and no-arbitrage in stochastic securities market models
cites work: On the different notions of arbitrage and existence of equilibrium
cites work: Q2725602
cites work: A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time
DOI: 10.1007/S10203-003-0039-5
description: scientific article; zbMATH DE number 2021365