We consider one-dimensional bedforms (e.g., anti-dunes, dunes, ripples), and investigate their stability and stochastic dynamics by combining numerical stochastic simulations and linear stability analysis. We focus on anti-dune development in supercritical flows on sloping gravel beds. We also discuss how theory can be extended to ripples and dunes in subcritical flows on sand beds. We developed a model consisting of the classic Saint-Venant-Exner (SVE) equations and a stochastic advection-diffusion equation for particle activity (the solid volume of particles in motion per unit streambed area). The model is solved numerically using a weighted essentially nonoscillatory finitevolume method. It is applied to two case studies: (i) particle activity fluctuation over fixed plane beds with exact analytical solutions and (ii) the nonlinear simulation of the ensemble-averaged SVE for uniform background flow. Simulations are in excellent agreement with theoretical solutions. Also, they successfully capture the antidune regime and wavelength observed experimentally. A third set of simulations shows that the stochastic approach performs better than deterministic ones (referred to as non-equilibrium sediment transport equations in the hydraulic literature) at capturing not only the average sediment transport rate, but also its standard deviation and higher-order moments. Finally, a benchmark analysis of different bedload transport models provides evidence that those including particle diffusion perform better than classic models.