Understanding the complete spectrum of vibration phenomena remains a theme of chronic effort in rotary drilling and vibration-assisted (VAD) technology due to the stochastic nature of bottom-hole assembly (BHA) dynamics, and the limited number of models involving probabilistic approaches. In particular, lateral vibration represents an aggressive and disruptive type of oscillatory pattern given its high frequency content and ability to induce geometrical variations, centrifugal-induced bowing patterns, and severe bore-hole damage. In this study, three improved mathematical representations are proposed with the intent of pragmatically characterizing the manifestation of phenomenological irregularities induced from bit-rock interference, fluid motion along the annulus, and recent VAD technology. In this manner, elucidating the complex physical attributes of a demanding engineering problem surrounding the national economy is achieved. Parameter identification for each dynamic model implies incorporating a finite element technique, where the flexibility of the drill-string and elastic characteristics of the well-bore are accounted for. To address the nature of the nonlinearity, the method of statistical linearization is incorporated to replace the nonlinear dynamical system with a set of linear equations, and thus establish an exact, analytical form of solution. Further, the stochastic nature of the BHA is addressed by imposing stationary/non-stationary excitations at the drill-bit segment and implementing Monte Carlo simulation to approximate the corresponding spectral density function. For this purpose, colored noise is filtered through an auto-regressive scheme to replicate the performance of a polycrystalline diamond compact (PDC) drill-bit.