Following Giovanni Volpe and Jan Wehr (2016 Rep. Prog. Phys. 79 053901), we analyze a minimal discrete-time dynamical system that models a colloid in a homogenous fluid (far from the boundary). In the presence of a boundary-induced diffusion gradient, the most obvious generalization of the model fails to exhibit the expected behavior. We show that this generalization approximates the Itô integral, while the correct dynamics of a near-equilibrium system are captured by the so-called "anti-Itô" model.