Utility Maximization with Habit Formation: Dynamic Programming and Stochastic PDEs

This paper studies 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, identify the associated value function as a generalized utility function, and exploit the interplay between dynamic programming and Feynman–Kac results via the theory of random fields and stochastic partial differential equations (SPDEs). The resulting value random field of the optimization problem satisfies a nonlinear, backward SPDE of parabolic type, widely referred to as the stochastic Hamilton–Jacobi–Bellman equation. The dual value random field is characterized further in terms of a backward parabolic SPDE which is linear. Progressively measurable versions of stochastic feedback formulae for the optimal portfolio and consumption choices are obtained as well.