Profitability in a multiple strategy market
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 \(\widetilde P\).
scientific article; zbMATH DE number 2021365