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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