Optimal Consumption and Portfolio Selection with Stochastic Differential Utility

We develop 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 D. Duffie and L. Epstein (1992, Econometrica 60, 353 394). We characterize the first-order conditions of optimality as a system of forward backward SDEs, which, in the Markovian case, reduces to a system of PDEs and forward only SDEs that is amenable to numerical computation. Another contribution is a proof of existence, uniqueness, and basic properties for a parametric class of homothetic SDUs that can be thought of as a continuoustime version of the CES Kreps Porteus utilities studied by L. Epstein and A. Zin (1989, Econometrica 57, 937 969). For this class, we derive closed-form solutions in terms of a single backward SDE (without imposing a Markovian structure). We conclude with several tractable concrete examples involving the type of ``affine'' state price dynamics that are familiar from the term structure literature. Journal of Economic Literature Classification Numbers: G11, E21, D91, D81, C61. 1999