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With zero interest and zero risk premium, a manager whose current wealth lies between common-tangent contacts $\ell ,h$ can improve expected utility maximization through a fair lottery paying those two values. The probability of $h$ is $(w_0-\ell )/(h-\ell )$. In a Brownian filtration, replicating a bounded terminal lottery gives a nonnegative wealth martingale throughout.

An investment-consumption problem chooses portfolio holdings and consumption to maximize discounted expected utility maximization subject to a self-financing portfolio wealth equation and admissibility constraints. A Hamilton-Jacobi-Bellman equation or utility duality with martingale deflators characterizes the optimum when the value is finite.