Quantum stochatic integrals and Doob-Meyer decomposition

Details Quantum stochatic integrals and Doob-Meyer decomposition Quantum stochatic integrals and Doob-Meyer decomposition

2006-02-10

arxiv.org/abs/math/0602216v1

Mathematics

Operator Algebras

34 pages

Scientific paper

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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