Foundations of continuous-time recursive utility: differentiability and normalization of certainty equivalents
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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.
KeywordsRecursive utility Stochastic differential utility Lévy framework Certainty equivalents Normalization Dynamic programming
JEL ClassificationD81 D91 C61
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- 2.Benzoni, L., Collin-Dufresne, P., Goldstein, R.: Can standard preferences explain the prices of out-of-the-money S&P 500 put options? NBER Working Paper W11861 (2005)Google Scholar
- 4.Bhamra, H.S., Kuehn, L.-A., Strebulaev, I.A.: The aggregate dynamics of capital structure and macroeconomic risk. Rev. Financ. Stud. (2010) (forthcoming)Google Scholar
- 6.Buckdahn, R., Pardoux, E.: BSDE’s with jumps and associated integro-partial differential equations, preprint (1994)Google Scholar
- 17.Fisher, M., Gilles, C.: Consumption and asset prices with recursive preferences. FEDS Working Paper 98–40 (1998)Google Scholar
- 27.Skiadas C.: Dynamic portfolio choice and risk aversion. In: Birge, J.R., Linetsky V., (eds) Handbooks in Operation Research and Management Science, vol. 15, pp. 789–843. Elsevier, New York (2008)Google Scholar
- 28.Skiadas, C.: Smooth ambiguity aversion toward small risks and continuous-time recursive utility, preprint. http://ssrn.com/abstract=1238156 (2008)