Aspects Of Utility Maximization With Habit Formation: Dynamic Programming And Stochastic PDE's

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 (SPDE’s). The resulting value random field of the optimization problem satisfies a non-linear, 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.

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