Supervised classification of diffusion paths

Let X = ( X t ) t ∈[0,1] be a stochastic process with label Y ∈ {0, 1}.We assume that X is some Brownian diffusion when Y = 0, while X is another Brownian diffusion when Y = 1. Based on an explicit computation of the Bayes rule, we construct an empirical classification rule drawn from an i.i.d. sample of copies of ( X , Y ). In a nonparametric setting, we prove that is a consistent rule, and we derive its rate of convergence under mild assumptions on the model.