Suppose I have some data which is assumed to follow a Ornstein-Uhlenbeck process defined by the SDE:
$$dX_t = \theta (\mu - X_t)dt + \sigma dW_t$$
According to Wikipedia and other sources I've found online this is a stationary Gaussian process, which only differs from normal Brownian motion with drift in that the drift is mean reverting instead of constant.
Now suppose I have sampled such a process at discrete points
$$X_0, X_\Delta, X_{2\Delta}, ..., X_{n\Delta}$$
I am interested in what I can say about the differences
$$\xi_i = X_{i\Delta} - X_{(i-1)\Delta}$$
Looking at the solution for $X_t$ given here, it seems clear that these are not independent, nor are they identically distributed. They are however Gaussian as the random components of their difference are two stochastic integrals with respect to a Wiener process.
So my question is, what exactly can be said about these differences? Is there any way they can be used to test the original Ornstein-Uhlenbeck process? Or is it a waste of time differencing the data and one should restrict oneself to the original data?