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We study consumption-portfolio and asset pricing frameworks with recursive preferences and unspanned risk. We show that in both cases, portfolio choice and asset pricing, the value function of the investor/representative agent can be characterized by a specific semilinear partial differential equation. To date, the solution to this equation has mostly been approximated by Campbell-Shiller techniques, without addressing general issues of existence and uniqueness. We develop a novel approach that rigorously constructs the solution by a fixed point argument. We prove that under regularity conditions a solution exists and establish a fast and accurate numerical method to solve consumption-portfolio and asset pricing problems with recursive preferences and unspanned risk. Our setting is not restricted to affine asset price dynamics. Numerical examples illustrate our approach.
We study consumption-portfolio and asset pricing frameworks with recursive preferences and unspanned risk. We show that in both cases, portfolio choice and asset pricing, the value function of the investor/ representative agent can be characterized by a specific semilinear partial differential equation. To date, the solution to this equation has mostly been approximated by Campbell-Shiller techniques, without addressing general issues of existence and uniqueness. We develop a novel approach that rigorously constructs the solution by a fixed point argument. We prove that under regularity conditions a solution exists and establish a fast and accurate numerical method to solve consumption-portfolio and asset pricing problems with recursive preferences and unspanned risk. Our setting is not restricted to affine asset price dynamics. Numerical examples illustrate our approach.
This paper relates recursive utility in continuous time to its discrete-time origins and provides a rigorous and intuitive alternative to a heuristic approach presented in [Duffie, Epstein 1992], who formally define recursive utility in continuous time via backward stochastic differential equations (stochastic differential utility). Furthermore, 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 expected utility type. Conversely, we prove that in general L´evy frameworks this is essentially also necessary, i.e. aggregators that are not of 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. JEL Classifications: D81, D91, C61