How to test that a given process is an Ornstein–Uhlenbeck process

We show asymptotic distributions of the residual process in Ornstein–Uhlenbeck model, when the model is true. These distributions are of Brownian motion and of Brownian bridge, depending on whether we estimate one parameter or two. This leads to seemingly simple asymptotic theory of goodness of fit tests based on this process. However, next we show that the residual process would lead to a deficient testing procedures, unless a transformed form of it is introduced. The transformed process is introduced and their role is explained through connection with what is known for the so called chimeric alternatives in testing problems for samples.