General Analytical Solution for Mertons Type Consumption-Investment Problems

We solve analytically the Merton's problem of an investor with time-additive power utility. For general state dynamics, we prove existence of two power series representations of the relevant optimal policies and value functions, which hold for all admissible risk aversion parameters. We characterize all terms in the power series by a recursive formula, allowing analytical computations to arbitrary order. Some applications to explicit model settings highlight a very satisfactory accuracy of finite order approximations provided by our power series solution approach.