Before revision: For nonlinear stochastic differential systems, we develop strong fully implicit positivity preserving numerical methods in the case that the zero boundary is non-attracting. These methods are implicit in the diffusion vector fields. They thus apply to a restricted class, namely those with sublinear form. This however, still includes most Langevin derived processes typical of volatility models in finance and molecular simulation in physics. When the  zero boundary is attracting and attainable, we specialize to a prototypical model, namely the mean-reverting Cox--Ingersoll--Ross process.
After revision: In the Heston stochastic volatility model, the zero boundary is attracting and attainable, we specialize to a prototypical model, namely the mean-reverting Cox--Ingersoll--Ross process.