Unbounded linear terminal-wealth utility ($\sup \{\mathbb{E}X:X\ge 0,\mathbb{E}[{\zeta }_{T}X]\le w_0\}=\infty$)

If a positive terminal state-price density has essential infimum zero, let $A_n=\{{\zeta }_T<1/n\}$ and $X_n=w_0{1}_{A_n}/\mathbb{E}[{\zeta }_T{1}_{A_n}]$. Whenever $\mathbb{P}(A_n)>0$, these claims cost $w_0$ and satisfy $\mathbb{E}{X}_n>n{w}_0$. 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.