= Transversality and fundamental dividend prices
{title2=$S_t=\zeta_t^{-1}\mathbb E_t\sum_{s>t}\zeta_sd_s$}
Iteration of the pricing Euler equation gives
$$
\zeta_tS_t=\mathbb E_t\left[\sum_{s=t+1}^N\zeta_sd_s+\zeta_NS_N\right].
$$
The displayed fundamental price follows when the future dividend series is integrable and $\mathbb E_t[\zeta_NS_N]\to0$. Without this <dividend-price transversality condition>, a deflated price <martingale> component can survive. For individual holdings, the analogous terminal term is $\mathbb E[\zeta_NS_N\theta_{N+1}]$. Telescoping the <discrete dividend budget equation> gives the initial wealth cost of <consumption> minus this terminal term; its vanishing is needed for budget equality.
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