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A Generalized Stochastic Differential Utility

University of British Columbia (Commerce School, Finance Department), 2053 Main Mall, Vancouver, British Columbia, Canada V6T172, and Université d'Evry, Avenue du Père Jarlan, 91025 Evry Cedex, France
Université de Marne la Vallée, Equipe d'analyse et de mathématiques appliquées, Cité Descartes, 5, Bd. Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France
lazrak{at}commerce.ubc.ca
quenez{at}math.univ-mlv.fr

This paper generalizes, in the setting of Brownian information, the Duffie–Epstein (1992) stochastic differential formulation of intertemporal recursive utility (SDU). We provide a utility functional of state-contingent consumption plans that exhibits a local dependency with respect to the utility intensity process (the integrand of the quadratic variation) and call it the generalized SDU. This mathematical generalization of the SDU permits, in fact, more flexibility in the separation between risk aversion and intertemporal substitution and allows to model asymmetry in risk aversion.

We extensively use the backward stochastic differential equation theory to give sufficient conditions for comparative and absolute risk aversion behavior as well as aversion to specific directional risk. Additionally, we discuss whether our functional exhibits monotonicity to its information filtration argument. For purposes of illustration, we provide some applications to the consumption/portfolio strategy selection problem in a complete securities market.

Key Words: Recursive utility; backward stochastic differential equations; risk aversion; uncertainty resolution; Brownian filtrations
History: Received: January 22, 2001; revision received: June 25, 2002;revision received: August 19, 2002;