This talk presents recent advances in numerical methods for stochastic models. In the first part, we introduce an adaptive splitting scheme for the Cox–Ingersoll–Ross (CIR) process that preserves positivity by dynamically adjusting step sizes and using a positivity-preserving backstop method. We establish convergence results and demonstrate the effectiveness of the scheme through simulations. In the second part, we discuss stochastic partial differential equations driven by fractional Brownian motion under different noise structures. Combining a Galerkin discretization in space with exponential Euler methods in time, we develop efficient schemes with  error estimates.  These results illustrate how numerical methods can address structural challenges in stochastic modeling, with applications in finance, physics, and beyond.