Recursive algorithm for transition density approximation and simulation of diffusion processes

Diffusion processes and more generally, stochastic differential equations (SDEs), are widely used to model natural and financial systems. However, accurately simulating them remains challenging due to the limitations of discretization methods. We propose a recursive algorithm to approximate the transition density of scalar diffusion processes using Hermite polynomial expansions. Unlike standard numerical schemes, our method uses an expansion in Hermite polynomials to approximate the transition density without requiring an arbitrarily small discretization step. This approximation is then used to simulate diffusion paths with high fidelity. Numerical experiments, including the Vasicek and CIR processes, confirm the effectiveness and efficiency of the method.