Different kinds of two-parameter martingales
Different kinds of two-parameter martingales were progressively introduced by several authors on a two-parameter stochastic basis (\(\Omega\),\({\mathcal F},P,({\mathcal F}_{st})_{s,t\geq 0}):\) weak martingales, i-martingales, strong martingales, martingales with orthogonal increments, path-invariant martingales and direction-invariant martingales. The purpose of this paper is to study the various relations between these concepts, supposing that the filtration satisfies the (F4)-hypothesis of Cairoli and Walsh. Denote these classes of martingales (supposing also that they are \(L^ 2\)-bounded) by \({\mathcal M}^ 2_ w\), \({\mathcal M}^ 2_ i\), \({\mathcal M}^ 2_ s\), \({\mathcal M}^ 2_ o\), \({\mathcal M}^ 2_{piv}\), \({\mathcal M}^ 2_{div}\), respectively. The main results of the paper are the following: 1. \({\mathcal M}^ 2_ s\subset {\mathcal M}^ 2_ o\), \({\mathcal M}^ 2_{div}={\mathcal M}^ 2_ o\cap {\mathcal M}^ 2_{piv}\) and there exist filtrations on which the inclusions are strict; 2. If the filtration is generated by a two-parameter Poisson process, then \({\mathcal M}^ 2_{div}={\mathcal M}^ 2_ s.\) Note also an interesting intermediary result: \(M\in {\mathcal M}^ 2_{piv}\) iff \(M^ 2-<M>\) is a martingale (in general \(M^ 2-<M>\) is only a weak martingale).
scientific article; zbMATH DE number 3969784
scientific article