Using the laws of probability, we can always factor such a joint density into a product of conditional densities:

$$

\begin{align}

\begin{aligned}

p(W_T, W_{T-1}, \ldots, W_1, W_0) = & p(W_T | W_{T-1}, \ldots, W_0) p(W_{T-1} | W_{T-2}, \ldots, W_0) \cdots \cr

& \quad \quad \cdots p(W_1 | W_0) p(W_0)

\end{align}

\end{aligned}

$$

In general,

$$

P_p =

\begin{bmatrix}

\frac{1}{2} & \frac{1}{2} \\

\frac{1}{4} & \frac{3}{4}

\frac{2}{3} & \frac{1}{3} \\

\frac{2}{3} & \frac{1}{3}

\end{bmatrix}

$$

time $t$ function can be expressed

$$

\begin{equation}

\begin{split}

\begin{aligned}

\pi_{t} =&p_{t}S_{t}-c\left(Q_{t}\right)-d\left(I_{t},S_{t}\right) \\

=&\left(a_{0}-a_{1}S_{t}+v_{t}\right)S_{t}-c_{1}Q_{t}-c_{2}Q_{t}^{2}-d_{1}I_{t}-d_{2}\left(S_{t}-I_{t}\right)^{2} \\

=&a_{0}S_{t}-a_{1}S_{t}^{2}+Gz_{t}S_{t}-c_{1}Q_{t}-c_{2}Q_{t}^{2}-d_{1}I_{t}-d_{2}S_{t}^{2}-d_{2}I_{t}^{2}+2d_{2}S_{t}I_{t} \\

I_{t}\\

z_{t}

\end{array}\right]\right)

\end{split}

\end{equation}

\end{aligned}

$$

where $S_{c}=\left[1,0\right]$.

are wrapped in a class called `LQ`, which includes

* Instance data:

* The required parameters $Q, R, A, B$ and optional parameters C, β, T, R_f, N specifying a given LQ model

* The required parameters $Q, R, A, B$ and optional parameters $C, \beta, T, R_f, N$ specifying a given LQ model

* set $T$ and $R_f$ to `None` in the infinite horizon case

* set `C = None` (or zero) in the deterministic case

* the value function and policy data