Quadratic variation for a class of L lo\(g^ +\,L\)-bounded two-parameter martingales
The authors prove that every L log\({}^+ L\)-bounded (not necessarily continuous) two-parameter martingale possesses a quadratic variation in \(L^ 0(\Omega,{\mathcal F},P)\) under the assumption that the one-parameter marginal filtrations \({\mathcal F}^ 1\) and \({\mathcal F}^ 2\) are quasi-left continuous. The technique used is a two-parameter Burkholder-type inequality for martingales indexed by \({\mathbb{N}}^ 2\) and truncation methods that reduce the L log\({}^+ L\)-bounded case to the bounded case. The authors also prove that the bounded martingale that has orthogonal increments or path- independent variation possesses quadratic variation (without hypothesis of quasi-left continuity of \({\mathcal F}^ 1\) and \({\mathcal F}^ 2)\).
two-parameter Burkholder- type inequality
scientific article; zbMATH DE number 4026464