Optimal consumption and portfolio selection with stochastic differential utility

We develop the utility gradient (or martingale) approach for computing port folio 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 t he Markovian case, reduces to a system of PDEs and forward only SDEs that i s amenable to numerical computation, Another contribution is a proof of exi stence, uniqueness, and basic properties for a parametric class of homothet ic SDUs that can be thought of as a continuous-time version of the CES Krep s-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 s ingle backward SDE (without imposing a Markovian structure), We conclude wi th several tractable concrete examples involving the type of "affine" state price dynamics that are familiar from the term structure literature, Journ al of Economic Literature Classification Numbers: G11, E21, D91, D81, C61. (C) 1999 Academic Press.