# Modelling EUR/USD rate with Ornstein-Uhlenbeck model

I have a data set of daily EUR/USD rate for time period 2000-2018. My goal is to model future behaviour of this financial time series using Ornstein-Uhlenbeck model: $$d X_t = \alpha (\theta - X_t) d t + \sigma d B_t$$ In order to calibrate the model I used MLE (Maximum Likelihood estimation) to obtain estimators for model parameters $\alpha, \theta$ and $\sigma$. Bellow are resulting estimators:

$$\hat{\alpha} = 0.00103 \\ \hat{\theta} = 1.27334 \\ \hat{\sigma} = 0.00647$$

Whereas estimated value for $\theta$ seems to be pretty reasonable (long-term mean to which the process tends to revert), value for $\alpha$ being close to $0$ does not make much sense to me. As a result the first mean-reverting part of the equation basically drops out and what is left is "weak" (due to small value of $\sigma$) Brownian motion $d X_t = \hat{\sigma} d B_t$. Consequently my simulated process is not modelling the real process at all as can be seen in the picture below: I am aware of the fact that this problem might be related to lack of stationarity in my data (p-value for ADF test was around 0.6, meaning we did not reject null hypothesis of non-stationarity).

Is this the only reason for poor modelling power of my calibrated model? If so, what are the best ways to tackle the problem of non-stationarity? Are there any other possible reasons for $\alpha$ being close to 0? Also, is O-U model even suitable for modelling exchange rate processes? If not, what models would be more approriate?