Brownian motion is a central scientific paradigm. Recently, due to increasing efforts and interests towards miniaturization and small-scale physics or biology, the effects of confinement on such a motion have become a key topic of investigation. Essentially, when confined near a wall, a particle moves much slower than in the bulk due to friction at the boundaries. The mobility is therefore locally hindered and space-dependent, which in turn leads to the apparition of so-called multiplicative noises. Here, we present efficient, broadrange and quantitative numerical simulations of a such a problem. Specifically, we integrate the overdamped Langevin equation governing the thermal dynamics of a negatively-buoyant spherical colloid within a viscous fluid confined by rigid walls, including surface charges. From the produced large set of long random trajectories, we perform a complete statistical analysis and extract all the key quantities, such as the probability distributions in displacements and their first moments. In particular, we propose a convenient method to compute high-order cumulants by reducing convergence problems, and employ it to characterize the inherent non-Gaussianity of the process.