Donati-Martin, C. Stochastic integration with respect to \(q\) Brownian motion. (English) Zbl 1033.60066 Probab. Theory Relat. Fields 125, No. 1, 77-95 (2003). 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. Cited in 2 ReviewsCited in 16 Documents MSC: 60H05 Stochastic integrals 81S25 Quantum stochastic calculus Keywords:\(q\)-Brownian motion; non-commutative stochastic processes; Fock space; quantum stochastic calculus; free probability Citations:Zbl 0873.60087 PDFBibTeX XMLCite \textit{C. Donati-Martin}, Probab. Theory Relat. Fields 125, No. 1, 77--95 (2003; Zbl 1033.60066) Full Text: DOI