A Generalized Stochastic Differential Utility

Abstract

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.

Risk- and ambiguity-averse portfolio optimization with quasiconcave utility functionals

“…For this choice, the problem is naturally formulated in terms of utility from consumption (or stochastic differential utilities) and the natural tool is the theory of BSDEs. While a systematic study was initiated in [75], these results have been considerably extended in a number of articles; see [8,15,26,40,56]. For the case of utility from terminal wealth and general variational criteria, stochastic control methods have been successfully applied for the choice of logarithmic utility, for stochastic factor models as well as for the non-Markovian case; see, respectively, [38,39] and [53].…”

Robust Portfolio Choice and Indifference Valuation

“…Recently, there has been a growing interest in the effects of ambiguity on portfolio choice and valuation; see, for example, Chen and Epstein [11], Lazrak and Quenez [48], Maenhout [52], Müller [59], Schied [70], Klöppel and Schweizer [45], Föllmer, Schied and Weber [28], Owari [61] and Sircar and Sturm [71]. The importance of incorporating ambiguity in the problems of portfolio choice and valuation is not merely theoretical as ambiguity plays a potential role in addressing important failures of purely risk-based settings that rule out model uncertainty.…”

“…BSDEs play an important role in stochastic control; see, for example, Pardoux and Peng [62], Duffie and Epstein [22], El Karoui, Peng and Quenez [23], Chen and Epstein [11], Lazrak and Quenez [48], Skiadas [72], Lim [49,50], Hamadène and Jeanblanc [33], Horst and Müller [36], and also the early work of Bismut [7]. In a Markovian setting, BSDEs correspond to semi-linear PDEs.…”

Robust Preferences and Robust Portfolio Choice

“…A direct BSDE approach to the primal problem was given by MÜLLER [2005]. Related problems arise in the maximization of recursive utilities in the sense of DUFFIE and EPSTEIN [1992]; see, e.g., EL KAROUI et al [2001], LAZRAK and QUENEZ [2003], and the references therein. For the general notion of a BSDE and its applications to finance, we refer to EL KAROUI et al [1997].…”