A generalized stochastic differential utility

This paper generalizes, in the setting of Brownian information, the Duffie–Epstein 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 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.