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Stochastic integration with respect to \(q\) Brownian motion. (English) Zbl 1033.60066

Summary: A \(q\)-Brownian motion is an example of a non-commutative stochastic process, i.e. an operator-valued process, as it arises in quantum stochastic calculus or free stochastic calculus. The introduced construction of the \(q\)-Wiener process for \(q\in[-1,1]\) is based on M. Bożejko, B. Kümmerer and R. Speicher [Commun. Math. Phys. 185, 129–154 (1997; Zbl 0873.60087)]. The author develops a stochastic integration of bi-processes with respect to a \(q\)-Brownian motion and derives an Itô formula. Finally, the \(q\)-Brownian motion is shown to have a predictable representation type property.

MSC:

60H05 Stochastic integrals
81S25 Quantum stochastic calculus

Citations:

Zbl 0873.60087
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