# What is the intuition behind the different information structures from the static to the continuous time ones?

It is a difficult and challenging problem at the same time, to model the information structure in theoretical models of economics and finance. The information structure in most of the literature is presented in the following three cases (as far as I'm concerned):

$$\textbf{Case I:}$$ (Static time economic models) The signal is a binary signal from the following set $$S=\{s_L,s_H\}$$ where $$L$$ stands for the low precision signal and $$H$$ for the high precision signal. The infromation structure is defined as $$I=\left(S,\mu\right)$$ where $$\mu:\Omega\to\Delta(S)$$ and $$\Omega$$ is the state space and to let us assume that $$\Omega=\{G,B\}$$ is also binary.

$$\textbf{Case II:}$$ (Static time economic models) Mostly in market microstructure information is presented as a noisy (private or public) signal $$\tilde{s}=\tilde{v}+\tilde{e}$$, where the vector $$\{\tilde{v},\tilde{e}\}$$ is jointly normally distributed; where $$\tilde{e}$$ is mean-zero i.i.d with variance $$\sigma_{e}^2$$ and $$\tilde{v}\sim N(v,\sigma_{v}^2)$$.

$$\textbf{Case III:}$$ (Continuous time economic models) Recently, I have seen some progress in the literature representing the information as a stochastic process that follows the dynamics of an SDE. For example, let us assume again that $$\theta$$ is the signal that captures the evolution of the state space $$\Theta$$. The initial state $$\theta_0$$ is normally distributed with mean $$\mu_0$$ and variance $$\sigma^2_0$$. The state then evolves according to the stochastic differential equation

$$d\theta_t=k(\delta-\theta_t)dt+\sigma dW_t\tag{OU}$$

where the driving process $$\{W_t\}_{t≥0}$$ is a standard Brownian motion, independent of the initial state $$\theta_0$$ and the variance $$\sigma^2$$ is strictly positive. The Ornstein-Uhlenbeck process satisfies the SDE $$(OU)$$ where the constants $$κ> 0$$ and $$\delta$$ are the rate of mean reversion and the long-term mean of the process respectively.

For any fixed $$s$$ and $$t$$, the random variable $$\theta_t$$, conditional upon $$\theta_s$$, is normally distributed with

$$(\hat{\mu},\hat{\sigma}^2)=\left(\delta+\left(\theta_s-\delta\right)e^{-k(t-s)},\frac{\sigma^2}{2k}\left(1-e^{-2k(t-s)}\right)\right)$$

$$\textbf{Question:}$$ Could we somehow evaluate which of the above cases is a better model to interpret the information structure in an economic environment? What is the intuition behind the use of each one of the information structures and taking into account the third case why an $$(OU)$$ is a better fit of any other stochastic processes?

• Should I re-post my question? Jan 19, 2023 at 9:31