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 continuous-time 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. (C) 1999 Academic Press.