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Mathematics and Financial Economics

, Volume 3, Issue 3–4, pp 115–138 | Cite as

Foundations of continuous-time recursive utility: differentiability and normalization of certainty equivalents

  • Holger KraftEmail author
  • Frank Thomas Seifried
Article

Abstract

This paper relates recursive utility in continuous time to its discrete-time origins and provides an alternative to the approach presented in Duffie and Epstein (Econometrica 60:353–394, 1992), who define recursive utility in continuous time via backward stochastic differential equations (stochastic differential utility). We show that the notion of Gâteaux differentiability of certainty equivalents used in their paper has to be replaced by a different concept. Our approach allows us to address the important issue of normalization of aggregators in non-Brownian settings. We show that normalization is always feasible if the certainty equivalent of the aggregator is of the expected utility type. Conversely, we prove that in general Lévy frameworks this is essentially also necessary, i.e. aggregators that are not of the expected utility type cannot be normalized in general. Besides, for these settings we clarify the relationship of our approach to stochastic differential utility and, finally, establish dynamic programming results.

Keywords

Recursive utility Stochastic differential utility Lévy framework Certainty equivalents Normalization Dynamic programming 

JEL Classification

D81 D91 C61 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of FinanceGoethe-UniversityFrankfurt am MainGermany
  2. 2.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany

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