In this exercise, we prove Proposition 9.8, considering a recursive preference functional of the form (9.19), with \(U_{T}(c, \theta)=u(c)=(1-\delta)^{1 /(1-ho)} c\).

(i) By following the arguments given in the proof of Proposition 6.4, show that the Euler conditions (9.23)-(9.24) hold.

(ii) Denoting by \(\left(W_{t}^{*}\right)_{t=0,1, \ldots, T}\) and \(\left(c_{t}^{*}\right)_{t=0,1, \ldots, T}\) the optimal wealth and consumption processes, respectively, in the maximization of the recursive preference functional \(U_{0}

(c, \theta)\) defined in (9.19), define the adapted stochastic process \(\left(\xi_{t}\right)_{t=0,1, \ldots, T}\) as the solution to the recursive relation

\[\xi_{t+1}:=\delta\left(\frac{V\left(W_{t+1}^{*}, t+1\right)}{\mu_{t}\left(V\left(W_{t+1}^{*}, t+1\right)\right)}\right)^{\varrho-\alpha}\left(\frac{c_{t+1}^{*}}{c_{t}^{*}}\right)^{-\varrho} \xi_{t} \quad \text { and } \quad \xi_{0}=1\]

Consider the backward stochastic difference equation

\[X_{t}=c_{t}^{*}+\mathbb{E}\left[\left.\frac{\xi_{t+1}}{\xi_{t}} X_{t+1} \rightvert\, \mathscr{F}_{t}\right], \quad \text { for } t=0,1, \ldots, T-1 \text {, with } X_{T}=\frac{V\left(W_{T}^{*}, T\right)}{u^{\prime}\left(c_{T}^{*}\right)},\]

the solution of which is an adapted stochastic process \(\left(X_{t}\right)_{t=0,1, \ldots, T}\). Show that this backward stochastic difference equation admits a unique solution which is given by \(X_{t}=W_{t}^{*}\), for all \(t=0,1, \ldots, T\). Note also that the process \(\left(\xi_{t}\right)_{t=0,1, \ldots, T}\) corresponds to the stochastic discount factor introduced in (9.29).
(iii) Deduce that the optimal wealth process \(\left(W_{t}^{*}\right)_{t=0,1, \ldots, T}\) satisfies relation (9.25) and that the return process \(\left(r_{t}^{*}\right)_{t=1, \ldots, T}\) associated to the optimal wealth process can be represented as in (9.26).
(iv) Deduce the validity of the Euler condition (9.27).