Utility consumption investment optimization in complete markets and applications in dynamic programming
A general consumption/ investment problem is considered for an agent whose actions cannot affect the market prices, and who strives to maximize total expected discounted utility of both consumption and/or terminal wealth. Under very general conditions on the nature of the market model and on the utility functions of the agent, it is shown how to approach the above problem by considering both simultaneously and separately the two more elementary ones of maximizing utility of consumption only and of maximizing utility of terminal wealth only, or even appropriately composing them. The optimal consumption and wealth processes are obtained quite explicitly. In the case of a market model with deterministic coefficients, the optimal portfolio and consumption rules are explicitly derived in feedback form on the current level of wealth. Furthermore, the Hamilton-Jacobi- Bellman equation of dynamic programming is developed for the value function of the above utility optimization problem. In contrast to this nonlinear partial differential equation which governs the value function, its dual value function turns out to satisfy a linear one. The Monte Carlo simulation method is used for numerical applications which aim to the computation of the value function for a wide range of initial endowments.