The main goal of the asymptotic equivalence theory of Le Cam (1986) is to approximate general statistical models by simple ones. We develop here a global asymptotic equivalence result for nonparametric drift estimation of a discretely observed diffusion process and its Euler scheme. The asymptotic equivalences are established by constructing explicit equivalence mappings. The impact of such asymptotic equivalence results is that it justifies the use in many applications of the Euler scheme instead of the diffusion process. We especially investigate the case of diffusions with non constant diffusion coefficient. To obtain asymptotic equivalence, experiments obtained by random change of times are introduced.