Optimal Consumption and Portfolio Selection with Stochastic Differential Utility

This paper develops the utility gradient (or martingale) approach for computing portfolio and consumption plans that maximize stochastic differential utility (SDU), a continuous-time version of recursive utility due to Duffie and Epstein (1992a). The setting is that of a general stochastic investment opportunity set with Brownian information (making some of the results novel in the time-additive case, as well). We characterize the first order conditions of optimality as a system of forward-backward SDE's, and for the Markovian case we show how to solve this system in terms of a system of quasilinear parabolic PDE's and forward only SDE's, which is amenable to numerical computation. Another contribution is a proof of existence, uniqueness, and basic properties for a parametric class of homothetic SDU that can be thought of as a continuous-time version of the CES Kreps-Porteus utilities studied by Epstein and Zin (1989). For this class, we show that the solution method simplifies significantly, resulting in closed form solutions in terms of a single backward SDE (without imposing a Markovian structure). The latter can be easily computed, as we will illustrate with a number of tractable concrete examples involving the type of ``affine'' state price dynamics that are familiar from the term structure literature.