Natural processes and Doob-Meyer decompositions over a probability gage space
The paper is devoted to a non-commutative generalization of natural processes in the sense of Meyer. In Section 1 the naturalness is reformulated so that it may be used in the non-commutative setting. In Section 2 probability gage spaces (\({\mathcal U},\tau)\) are considered where \({\mathcal U}\) is a finite von Neumann algebra, \(\tau\) is a faithful normal probability trace on \({\mathcal U}\). In this setting natural processes are introduced and some examples are given. This enables the authors to prove that Doob-Meyer decomposition obtained earlier by them and \textit{R. F. Streater} [ibid. 48, 172-212 (1982; Zbl 0492.46051)] is unique. Here they also introduce Nelson's ''regular processes'' in this non-commutative setting and show that \((| X_ t|^ 2)\) is a regular process for a wide class of martingales \((X_ t)\). In Section 3 the results of the previous sections are applied in the case of the Clifford probability gage spaces.
non-commutative generalization of natural processes
scientific article; zbMATH DE number 3879543
scientific article