In model predictive control, a high quality of control can only be achieved, if the model of the system reflects the real-world process as precisely as possible. Therefore, the controller should be capable of both handling a nonlinear system description and systematically incorporating uncertainties affecting the system. Since stochastic nonlinear model predictive control (SNMPC) problems in general cannot be solved in closed form, either the system model or the occurring densities have to be approximated. In this paper, we present an SNMPC framework, which approximates the densities and the reward function by their wavelet expansions. Due to the few requirements on the shape and family of the densities or reward function, the presented technique can be applied to a large class of SNMPC problems. For accelerating the optimization, we additionally present a novel thresholding technique, the so-called dynamic thresholding, which neglects coefficients that are insignificant, while at the same time guaranteeing that the optimal control input is still chosen. The capabilities of the proposed approach are demonstrated by simulations with a path planning scenario.