Under the assumptions of Sect. 6.4, prove that the following asset pricing relation holds true, for all \(n=0,1, \ldots, N\) and \(t=0,1, \ldots, T-1\) :

\[s_{t}^{n}=\sum_{s=1}^{T-t} \frac{\mathbb{E}\left[d_{t+s}^{n} \mid \mathscr{F}_{t}\right]}{r_{f}^{s}}+\sum_{s=1}^{T-t} \operatorname{Cov}\left(\delta^{s} \frac{\mathbf{u}^{\prime}\left(e_{t+s}\right)}{\mathbf{u}^{\prime}\left(e_{t}\right)}, d_{t+s}^{n} \mid \mathscr{F}_{t}\right)\]