Taking into account an old post of maths.stackexchange, I recall the following:
On the one hand, we know that the Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time AR(1) process.
An Ornstein–Uhlenbeck process, $s_t$, satisfies the following stochastic differential equation:
$$ ds_t = \theta (\mu-s_t)\,dt + \sigma\, dZ_t \tag{1}$$
where $(Z_t)_{(t\geq 0)}$ is the standard Wiener (or Brownian) process on a filtered probability space $\left(\Omega, \mathcal{F},(\mathcal{F}_t)_{(t\geq 0)},\mathbb{P}\right)$ and $\mathcal{F}_t$ is being the filtration generated by $Z_t$. Also, note that $\theta > 0, \mu$ and $\sigma > 0$ are $\mathcal{F}_t-$adapted.
The $AR(p)$ model, i.e. an autoregressive model of order $p$, is defined as
$$ X_t = c + \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon_t \, \tag{*}$$
where $\varphi_1, \ldots, \varphi_p$ are the parameters of the model, $c$ is a constant, and $\varepsilon_t$ is white noise.
Contrarily, in the Kyle seminal model, the private information of an informed trader is written as
$$s=v+\epsilon\tag{2}$$
where $v\sim N(\bar{V}, \sigma_V^2)$ and $\epsilon\sim N(0, \sigma_{\epsilon}^2) $ and $\epsilon$ is independently normally distributed. Note that, Kyles model is static.
My questions are the following.
- Could we find a continuous time analogue for the private information model in $(2)$ and could in be an SDE?
- If we could extend the Kyle model in a continuous analogue, under which circumstances $(2)$ could be modelled as an Ornstein–Uhlenbeck process which is defined as in (1)?
- If Ornstein–Uhlenbeck process is not suitable to model the dynamics of the private information of Kyle in (2) in a continuous time case, what would it be a suitable model?
