= Logarithmic terminal wealth in a complete market
{title2=$u(x)=\log x-\mathbb E\log Z$}
In a <complete market> with a unit cash asset and normalized strictly positive <state-price density> $Z$, the optimal positive terminal <portfolio wealth> is $x/Z$ whenever it is replicable in the admissible class. The <state-price budget constraint> is an equality: $\mathbb E[Z(x/Z)]=x$. The <utility conjugate> inequality gives $\mathbb E\log H\leq\log x-\mathbb E\log Z$ for every affordable positive claim. This value can be infinite, because $\mathbb E(\log Z)^+<\infty$ while $\mathbb E(\log Z)^-$ need not be finite. Even completeness for bounded claims suffices to approximate the value: set $b_n=\mathbb E[Z/(Z\vee n^{-1})]$ and $H_n=x/[b_n(Z\vee n^{-1})]$. Each claim is bounded and costs $x$, while its expected <logarithmic utility> tends to the displayed value by <monotone convergence> after an integrable lower bound is subtracted.
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