| This paper studies in detail the habit-forming preference problem of maximizing total expected utility from consumption net of the standard of living, a weighted-average of past consumption. We describe the effective state space of the corresponding optimal wealth and standard of living processes, while the associated value function is identified as a generalized utility function. In the case of deterministic coefficients, we exploit the interplay between dynamic programming and Feynman-Kac results, to obtain equivalent optimality conditions for the value function and its convex dual in terms of appropriate partial differential equations (PDE's) of parabolic type. The optimal portfolio/consumption pair is provided in feedback form as well.; In a more general context with random coefficients, this interrelation is established via the theory of random fields and stochastic PDE's. In fact, the resulting value random field of the optimization problem satisfies a non-linear, backward stochastic PDE of parabolic type, widely referred to as the stochastic Hamilton-Jacobi-Bellman equation. In addition, the dual value random field is characterized in terms of a linear, backward parabolic stochastic PDE. Employing the generalized Ito-Kunita-Wentzell formula, we present adapted versions of stochastic Feynman-Kac formulae, which lead to the formulation of stochastic feedback forms for the optimal portfolio and consumption choices. |
