# Journal of Operator Theory

Volume 66, Issue 1, Summer 2011 pp. 161-192.

Quantum stochastic integrals and Doob-Meyer decomposition**Authors**: Andrzej Luczak

**Author institution:**(1) Faculty of Mathematics and Computer Science, Lodz University, ul. S. Banacha 22, 90-238 Lodz, Poland

**Summary:**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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