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We consider a portfolio optimization problem which is formulated as a stochastic control problem. Risky asset prices obey a logarithmic Brownian motion, and interest rates vary according to an ergodic Markov diffusion process. The goal is to choose optimal investment and consumption policies to maximize the infinite horizon expected discounted hyperbolic absolute risk aversion (HARA) utility of consumption. A dynamic programming principle is used to derive the dynamic programming equation (DPE). The subsolution--supersolution method is used to obtain existence of solutions of the DPE. The solutions are then used to derive the optimal investment and consumption policies.
Fleming et al. (Thu,) studied this question.
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