I'm looking for a quick sanity check for something that is confusing me. According to Wikipedia if $X_t$ is an Ito process that can be written as a stochastic integral + a time integral then the term of finite variation in the decomposition $X_t = M_t + A_t$ is the time integral, and the martingale part $M_t$ is the stochastic integral.
If $X_t$ is a Ornstein Uhlenbeck process then it has a closed form $X_t = \int_0^t\sigma e^{-\theta(t-s)}dB_s$, and following this logic $A_t = 0$ and $X_t$ decomposes as a purely a martingale, but this is not possible if $X_t$ satisfies the SDE
$$dX_t = -aX_tdt + \sigma dB_t$$
Since solutions of an SDE are only martingales if they have no drift (here). What is the mistake I'm making to get this supposed contradiction?