  Following the approach and the terminology introduced in [A. Deya and R.
Schott, On the rough paths approach to non-commutative stochastic calculus, J.
Funct. Anal., 2013], we construct a product L{\'e}vy 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 detailled
comparison between the resulting "rough" integral and the stochastic "It{\^o}"
integral exhibited by Donati-Martin in [C. Donati-Martin, Stochastic
integration with respect to $q$ Brownian motion, Probab. Theory Related Fields,
2003].
