  We construct quadratic stochastic processes (QSP) (also known as Markov
processes of cubic matrices) in continuous and discrete times. These are
dynamical systems given by (a fixed type, called $\sigma$) stochastic cubic
matrices satisfying an analogue of Kolmogorov-Chapman equation (KCE) with
respect to a fixed multiplications (called $\mu$) between cubic matrices. The
existence of a stochastic (at each time) solution to the KCE provides the
existence of a QSP called a QSP of type $(\sigma | \mu)$. In this paper, our
aim is to construct and study trajectories of QSPs for specially chosen notions
of stochastic cubic matrices and a wide class of multiplications of such
matrices (known as Maksimov's multiplications).
