Stochastic approximation finite element method: Analytical formulas for multidimensional diffusion process

We derive an analytical weak approximation of a multidimensional diffusion process as coefficients are small or time is small. Our methodology combines the use of Gaussian proxies to approximate the law of the diffusion and a finite element interpolation of the terminal function applied to the diffusion. We call this the stochastic approximation finite element (SAFE) method. We provide error bounds of our global approximation depending on the diffusion process coefficients, the time horizon, and the regularity of the terminal function. Then we give estimates of the computational cost of our algorithm. This shows an improved efficiency compared to Monte-Carlo methods in small and medium dimensions (smaller than 10), which is confirmed by numerical experiments.