We provide sufficient conditions for the existence of a solution to a consumption and portfolio problem in continuous time under uncertainty with an infinite horizon. When the price processes for securities are diffusion processes, optimal policies can be computed by solving a linear partial differential equation. We also provide conditions under which the solution to an infinite horizon problem is the limit of the solutions to finite horizon problems when the horizon increases to infinity.
"Optimal Consumption and Portfolio Policies with an Infinite Horizon: Existence and Convergence." Ann. Appl. Probab. 2 (1) 36 - 64, February, 1992. https://doi.org/10.1214/aoap/1177005770