12.1

[luigift]

I've used slightly different approach. If you would like to incorporate the answer I can send you the LaTeX code.

[LyWangPX]

Yes, if you could provide the latex code I will incoroporate your answers and note your name in pdf.

[luigift]

I'll post it here. I recommend adding the latex files and latex generation instructions to the repository so that other people can open pull requests directly. In the future, adding github actions to build the pdf automatically would also be nice.

[luigift]

\begin{equation} \label{eq1} G{t:t+h} \doteq R{t+1} + \gamma G_{t+1:t+h} \end{equation}

\begin{equation} \label{eq2} G^\lambda{t+1} \doteq (1-\lambda) \sum^{\infty}{n=1} \lambda^{n-1}G_{t+1:t+n+1} \end{equation}

\begin{equation} \label{eq3} G{t+1:t+1} \doteq \hat{v}(S{t+1}, w_{t}) \end{equation}

\begin{align} \label{eq4} G^\lambdat & \doteq (1-\lambda) \sum^{\infty}{n=1} \lambda^{n-1}G{t:t+n} \ & = (1-\lambda) \left[G{t:t+1} + \lambda G{t:t+2} + \lambda^2 G{t:t+3} + \dots \right] \ & = (1-\lambda) \left[(R{t+1} + \gamma G{t+1:t+1}) + \lambda (R{t+1} + \gamma G{t+1:t+2}) + \lambda^2 (R{t+1} + \gamma G{t+1:t+3})+ \dots \right] & (\text{by \ref{eq1}})\ & = (1-\lambda) \left[(R{t+1} + \lambda R{t+1} + \lambda^2 R{t+1} + \dots) + (\gamma G{t+1:t+1} + \lambda \gamma G{t+1:t+2} + \lambda^2 \gamma G{t+1:t+3} + \dots) \right] \ & = (1-\lambda) \left[\frac{R{t+1}}{(1-\lambda)} + \gamma G{t+1:t+1} + \gamma(\lambda G{t+1:t+2} + \lambda^2 G{t+1:t+3} + \dots)\right] \ & = R{t+1} + (1-\lambda) \left[ \gamma G{t+1:t+1} + \gamma \lambda (G{t+1:t+2} + \lambda G{t+1:t+3} + \dots) \right] \ & = R{t+1} + (1-\lambda)\gamma G{t+1:t+1} + \gamma \lambda (1-\lambda) \sum^{\infty}{n=1} \lambda^{n-1}G{t+1:t+n+1} \ & = R{t+1} + (1-\lambda)\gamma G{t+1:t+1} + \gamma \lambda G^\lambda{t+1} & (\text{by \ref{eq2}})\ & = R{t+1} + (1-\lambda)\gamma \hat{v}(S{t+1}, w{t}) + \gamma \lambda G^\lambda_{t+1} & (\text{by \ref{eq3}})\ \end{align}

[luigift]

Thank you for mentioning my name. How are your interviews going?

[LyWangPX]

Thanks for your suggestions and update. Well, not so well if say so, but life goes on :)

[LyWangPX]

Uploaded and Fixed. Also update your info in readme. Thanks for your contributions. :)