On stochastic calculus with respect to q-Brownian motion

Following the approach and the terminology introduced in Deya and Schott (2013) 6], we construct a product Lévy area above the q-Brownian motion (for $q\in 0,1)$) and use this object to study differential equations driven by the process.We also provide a detailed comparison between the resulting “rough” integral and the stochastic “Itô” integral exhibited by Donati-Martin (2003) 7].