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


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.

  1. Could we find a continuous time analogue for the private information model in $(2)$ and could in be an SDE?
  2. 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)?
  3. 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?


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