= Unbounded linear terminal-wealth utility
{title2=$\sup\{\mathbb EX:X\geq0,\ \mathbb E[\zeta_TX]\leq w_0\}=\infty$}
If a positive terminal <state-price density> has essential infimum zero, let $A_n=\{\zeta_T<1/n\}$ and $X_n=w_0\mathbf1_{A_n}/\mathbb E[\zeta_T\mathbf1_{A_n}]$. Whenever $\mathbb P(A_n)>0$, these claims cost $w_0$ and satisfy $\mathbb EX_n>nw_0$. In a Brownian <complete market> with constant nonzero <market price of risk>, $\zeta_T$ has a <lognormal distribution> and all these events have positive probability. Thus bounded market coefficients and an increasing <concave> utility do not alone ensure a finite attained optimum: the linear utility $U(x)=x$ already fails. Usual inverse-marginal-utility formulas require further utility and integrability hypotheses.
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