Discretization Scheme(s) of a Brownian Diffusion

This chapter is devoted to the discretization schemes of the solutions of a stochastic differential equation driven by a Brownian motion (diffusion): the (discrete time and continuous) Euler scheme and the Milstein scheme. The existence of moments, the strong (or pathwise) convergence rate of both schemes are established under Lipschitz assumptions of the diffusion coefficients (Euler scheme) or of their partial derivatives (Milstein scheme). Several other important properties of these schemes are established (e.g., conditions for the simulability of the Milstein scheme in higher dimension, Lipschitz property of the flow, etc). The main weak error results for the Euler scheme, either under smoothness (Talay–Tubaro) or ellipticity (Bally–Talay) assumptions, are stated, with a detailed proof in the first setting. Applications to the Richardson-Romberg extrapolation to reduce the bias in Monte Carlo simulations is presented and illustrated in an example.