Quantum stochatic integrals and Doob-Meyer decomposition

2006/02/10 by Andrzej Luczak, Andrzej Łuczak, Luczak, Andrzej
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #FOS: Mathematics #Operator Algebras (math.OA) #Primary: 81S25 #Probability (math.PR) #Quantum Mechanics and Applications #Random Matrices and Applications #Secondary: 46L53 #math.OA #math.PR #msc:46L53 #msc:81S25

paper · pdf · doi:10.48550/arxiv.math/0602216

34 pages

arxiv created 2006/02/10 · openalex publication_date 2006/02/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that for a quantum L^p-martingale (X(t)), p>2, there exists a Doob-Meyer decomposition of the submartingale (|X(t)|²). A noncommutative counterpart of a classical process continuous with probability one is introduced, and a quantum stochastic integral of such a process with respect to an L^p-martingale, p>2, is constructed. Using this construction, the uniqueness of the Doob-Meyer decomposition for a quantum martingale `continuous with probability one' is proved, and explicit forms of this decomposition and the quadratic variation process for such a martingale are obtained.

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