Stochastic Differential Utility

Abstract

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.

Optimal Consumption and Portfolio Selection with Stochastic Differential Utility

“…The combined results of Duffie and Epstein [12] and Duffie and Skiadas [20] show that conditions C2 and C3 (with C=D + and V=D) hold if f satisfies certain continuity-Lipschitz-growth type conditions. 5 Moreover, 76 SCHRODER AND SKIADAS under the same restrictions on f, V 0 is continuous, V 0 is increasing if f is increasing in its consumption argument, and V 0 is concave if f is concave jointly in its consumption and utility arguments.…”

“…SDU was introduced by Duffie and Epstein [12] as a continuous time limit of the type of recursive utility studied by Kreps and Porteus [35], Epstein and Zin [24], and many others.…”

“…Moreover, they obtained closed-form expressions for marginal utilities in this sense for various types of temporally dependent preferences used in practice, including SDU. Even earlier, Duffie and Epstein [12] 69 OPTIMAL PORTFOLIO SELECTION used dynamic programming methods to compute the form of Arrow Debreu prices under SDU in a more special Markovian setting under Brownian information. The Duffie Epstein Skiadas papers, however, are oriented toward obtaining equilibrium pricing formulas and do not in fact solve the first-order conditions for the optimal consumption plan.…”

“…In addition to providing a general solution method, we will apply this approach to a parametric class of homothetic SDU that was introduced by Duffie and Epstein [12,13] as a continuous-time limit of the CES Kreps Porteus specification studied by Epstein and Zin [24] and by Weil [51]. Without Markovian assumptions, we will show that the first-order conditions simplify significantly in this case, resulting in closed-form expressions in terms of the solution to a single backward SDE.…”

Capital Taxation with Entrepreneurial Risk

“…For a more general proof of the above and for a proof of existence and uniqueness of the solution to the integral equation (31) see Duffie and Epstein (1992).…”