Optimal Consumption and Portfolio Selection with Stochastic Differential Utility, Journal of Economic Theory
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.
Mark Schroder, Constantinos Skiadas
Schroder, Mark, and Constantinos Skiadas. 1999. Optimal Consumption and Portfolio Selection with Stochastic Differential Utility. Journal of Economic Theory. 89(1): 68-126.