Stochastic Differential Utility and Asset Pricing

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 a “smooth” Markov setting, it is shown that utility can be represented as a smooth function of the underlying Markov state process, yielding a generalization of the Hamilton-Bellman-Jacobi characterization of optimality. As an application, homothetic representative-agent recursive utility functions are shown to imply that excess expected rates of return on securities are given by a linear combination of the continuous-time market-portfolio based CAPM and the consumption-based CAPM. The Cox, Ingersoll, and Ross characterization of the term structure is examined with a recursive generalization, showing the response of the term structure to variations in risk aversion. The paper also introduces a new multi-commodity factor-return model, as well as an extension of the “usual” discounted expected value formula for asset prices.