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!
Best Answer
In case $p=1$ you have $$ x_{k+1} = c+a x_k + b\varepsilon_k $$ so that if you put $c = \theta\mu\Delta t$, $a = -\theta\Delta t$ and $b = \sigma\sqrt{\Delta t}$ you get $$ x_{k+1} = \theta(\mu - x_k)\Delta t + \sigma \varepsilon_k\sqrt{\Delta t} $$ which is exactly an Euler-Maryuama discretization of OU at times $(k\Delta t)_{k\in \Bbb N_0}$.