This paper by Cox and Huang (1986) addresses the optimal consumption and portfolio policies in continuous time under uncertainty, focusing on the case where asset prices follow a diffusion process. The authors use a martingale representation technique to show the existence of optimal solutions without requiring compactness assumptions on admissible controls, which is a significant improvement over traditional stochastic control methods. They provide explicit characterizations of optimal consumption and portfolio policies through two main theorems and a verification theorem, which are counterparts to the verification theorem in dynamic programming. The approach is advantageous because it only requires solving a linear partial differential equation, unlike the nonlinear equations typically needed in dynamic programming. The paper also discusses the relationship between solutions with and without nonnegativity constraints on consumption and final wealth, and provides a special case analysis for geometric Brownian motion. The authors conclude by highlighting the practical implications of their findings for financial decision-making.This paper by Cox and Huang (1986) addresses the optimal consumption and portfolio policies in continuous time under uncertainty, focusing on the case where asset prices follow a diffusion process. The authors use a martingale representation technique to show the existence of optimal solutions without requiring compactness assumptions on admissible controls, which is a significant improvement over traditional stochastic control methods. They provide explicit characterizations of optimal consumption and portfolio policies through two main theorems and a verification theorem, which are counterparts to the verification theorem in dynamic programming. The approach is advantageous because it only requires solving a linear partial differential equation, unlike the nonlinear equations typically needed in dynamic programming. The paper also discusses the relationship between solutions with and without nonnegativity constraints on consumption and final wealth, and provides a special case analysis for geometric Brownian motion. The authors conclude by highlighting the practical implications of their findings for financial decision-making.