In this work, we approximate a diffusion process by its Euler scheme and we study the convergence of the density of the marginal laws. We improve previous estimates especially for small time.
Gobet, E., & Labart, C. (2008). Sharp estimates for the convergence of the density of the euler scheme in small time. Electronic Communications in Probability, 13, 352–363. https://doi.org/10.1214/ECP.v13-1393