Is Ornstein–Uhlenbeck process the continuous-time correspondence of AR(1) process?

Question

I see the AR(1) process (with $|\alpha| < 1$) can be written in the following way: $$x_{t+1} = \alpha x_t + \epsilon_t$$ $$\Delta x_t = - (1 - \alpha) x_t + \epsilon_t$$ which looks quite like the formula of Ornstein–Uhlenbeck process without a drift term as $$dx_t = -\theta x_t + \sigma dW_t$$ then is OU process the continuous-time correspondence of AR(1) process?

Ignore the question below if it is not. If I have the value of parameters of AR(1) process, i.e. $\alpha$, variance of $\epsilon$, and time difference $\Delta_t$, how do I convert these parameters to the parameter of OU process?

Answer 1

This link looks very relevant to your question and probably an answer.

https://math.stackexchange.com/questions/345773/how-the-ornstein-uhlenbeck-process-can-be-considered-as-the-continuous-time-anal