This paper presents a stochastic differential formulation of recursive utility. Sufficient conditions are given for existence, uniqueness, time consistency, monotonicity, continuity, risk aversion, concavity, and other properties. In the setting of Brownian information, recursive and intertemporal expected utility functions are observationally distinguishable. However, one cannot distinguish between a number of non-expected-utility theories of one-shot choice under uncertainty after they are suitably integrated into an intertemporal framework. In a "smooth" Markov setting, the stochastic differential utility model produces a generalization of the Hamilton-Jacobi-Bellman characterization of optimality. A companion paper explores the implications for asset prices.

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.

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