On martingale measures for stochastic processes with discrete time (Q2772019)

Let \(X(1), X(2),\dots\) be a sequence of bounded random variables defined on some probability space \((\Omega,{\mathcal F},P)\). Suppose that \({\mathcal F}= \sigma \{X(0), X(1),\dots\}\) \((X(0)=0)\). Let the set \(X[\Omega]= \{X(t)(\omega): t=1,2\dots\); \(\omega\in \Omega\}\) be closed in the Tikhonov topology in \(R^{\{1,2,\dots\}}\). Let the filtration be given by \({\mathcal F}_t= \sigma \{X(0),\dots, X(t)\}\), \(t= 0,1,\dots\) The authors show that under these assumptions the following two conditions are equivalent: NEWLINENEWLINENEWLINE(I) There exists a probability measure \(P^*\) on \({\mathcal F}\) such that \((X(t))\) is a \(P^*\)-martingale, and \(P^*|{\mathcal F}_t\sim P|{\mathcal F}_t\), \(t= 0,1,\dots\), NEWLINENEWLINENEWLINE(II) \(P(A\cap \{X(t+1)> X(t)\})>0 \iff P(A\cap \{X(t+1)< X(t)\})> 0\) for any \(A\in {\mathcal F}_t\) and \(t= 0,1,\dots\) NEWLINENEWLINENEWLINEThe authors do not discuss how condition (II) is related to certain (very general) conditions studied in a paper due to \textit{W. Schachermayer} [Math. Finance 4, 25-55 (1994; Zbl 0893.90017)].