How to test that a given process is an Ornstein-Uhlenbeck process (Q2046298)

The author considers the Ornstein-Uhlenbeck process described by the stochastic differential equation \(dX_t=\alpha(\mu-X_t)dt+dW_t\), \(t \geq 0\), \(X_0=0\) where \(W\) is the standard Wiener process. Here \(\alpha >0\) and \(\mu \in (-\infty,\infty)\). The author investigates the properties of the residual process when the parameters \(\alpha\) and \(\mu\) are estimated and it was shown that the asymptootic distribution of the residual process is a Brownian motion or a Brownian bridge depending on whether one parameter is estimated or both. It was further shown that the residual process will lead to a deficient testing procedure unless a transformed form of it is introduced. The author introduces the transformed process and studies its properties.