TITLE: How the Ornstein–Uhlenbeck process can be considered as the continuous-time analogue of the discrete-time AR(1) process?
QUESTION [36 upvotes]: Wikipedia says

The Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time AR(1) process.

I was wondering how the Ornstein–Uhlenbeck process can be considered as the continuous-time analogue of the discrete-time AR(1) process?
An Ornstein–Uhlenbeck process, $x_t$, satisfies the following stochastic differential equation:
$$
    dx_t = \theta (\mu-x_t)\,dt + \sigma\, dW_t
$$
where $\theta > 0, \mu$ and $\sigma > 0$ are parameters and $W_t$ denotes the Wiener process.
The $AR(p)$ model, i.e. an autoregressive model of order $p$, is defined as
$$
    X_t = c + \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon_t \,
$$
where $\varphi_1, \ldots, \varphi_p$ are the parameters of the model, $c$ is a constant, and $\varepsilon_t$ is white noise. 
Thanks and regards!

REPLY [22 votes]: Rather than using an approximation (like Euler-Maryuama), one can just sample* the continuous-time solution:
$$x(t)=x(0)\,e^{-\theta t}+\mu \,(1-e^{-\theta t})+\sigma \int _{0}^{t}e^{-\theta (t-s)}\,dW_{s}.\,
$$
Writing $X_k=x(k\,\Delta t)$, the sampled process is described by the AR(1) process:
$$
  X_{k+1} = c + \varphi \,X_k + \varepsilon_k
$$
where:
\begin{align}
X_0&=x(0),\\
\varphi &= e^{-\theta \Delta t},\\
c &= (1-\varphi)\mu, \\
\epsilon_k &\sim \mathcal{N}\Big(0,\frac{1}{2\theta}\sigma^2\big(1-e^{-2 \theta \Delta t}\big)\Big).
\end{align}
* I mean sample, not sample