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

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$$

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\begin{align}

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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

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& p(W_1 | W_0) p(W_0)

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\end{align}

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$$

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In general,

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time $t$ function can be expressed

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$$

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\begin{equation}

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\begin{split}

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

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=&\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} \\

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=&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} \\

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I_{t}\\

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z_{t}

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\end{array}\right]\right)

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\end{split}

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\end{equation}

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$$

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

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