On local comparability among families of sub-\(\sigma\)-fields (Q1122868)

The authors discuss the notion of local comparability in the case of m parameters \((m>2)\) and its application to multiparameter stochastic processes. Local comparability in the case of two parameters means that for two families of \(\sigma\)-fields \(\{\) \({\mathcal F}^ 1_ s\}_{s\in R_+}\) and \(\{\) \({\mathcal F}^ 2_ t\}_{t\in R_+}\) there exists \(A_{st}\in {\mathcal F}^ 1_ s\cap {\mathcal F}^ 2_ t\) such that \[ {\mathcal F}^ 1_ s|_{A_{st}}\subset {\mathcal F}^ 2_ t|_{A_{st}}\quad and\quad {\mathcal F}^ 2_ t|_{A^ c_{st}}\subset {\mathcal F}^ 1_ s|_{A^ c_{st}},\quad (s,t)\in {\mathbb{R}}^ 2_+. \] The authors prove that if m families of sub- \(\sigma\)-fields are locally comparable in pairs then they are mutually locally comparable. Each m-parameter family of sub-\(\sigma\)-fields which is locally comparable in pairs is determined by a single parameter family of sub-\(\sigma\)-fields and all martingales are strong. The properties of local comparability when the parameter set is a Riesz space are also considered.