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Finance and Stochastics

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12.04.2016

Consumption–investment optimization with Epstein–Zin utility in incomplete markets

verfasst von: Hao Xing

Erschienen in: Finance and Stochastics | Ausgabe 1/2017

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Abstract

In a market with stochastic investment opportunities, we study an optimal consumption–investment problem for an agent with recursive utility of Epstein–Zin type. Focusing on the empirically relevant specification where both risk aversion and elasticity of intertemporal substitution are in excess of one, we characterize optimal consumption and investment strategies via backward stochastic differential equations. The superdifferential of indirect utility is also obtained, meeting demands from applications in which Epstein–Zin utilities were used to resolve several asset pricing puzzles. The empirically relevant utility specification introduces difficulties to the optimization problem due to the fact that the Epstein–Zin aggregator is neither Lipschitz nor jointly concave in all its variables.

Vorheriger Artikel Optimal consumption and investment with Epstein–Zin recursive utility

Nächster Artikel Market completion with derivative securities

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The parameter \(1+\alpha\) in [38] is \(\theta\) here. Hence equation (8c) therein implies \(\theta>0\).

The specification \(\gamma,\psi>1\) is related to [38, Case 3 on page 113], which established the utility gradient inequality. Even though its proof is independent of the market model, it uses the existence and concavity of Epstein–Zin utility, which are established in [38, Appendix A] under the assumption \(\theta>0\). Therefore, one needs to replace [38, Appendix A] by Propositions 2.2 and 2.4 below which confirm the existence and concavity of Epstein–Zin utility when \(\theta<0\). During the revision of this paper, these properties were also confirmed in [40] for a general semimartingale setting.

This admissible set is similar to its counterpart in [11] for time-separable utilities, but is larger than its analogue in [38, Eq. (8a)], where \(\mathbb {E}[\int_{0}^{T} c_{s}^{\ell}ds]<\infty\) for all \(\ell\in \mathbb {R}\) is needed for an admissible consumption stream \(c\).

\(f\) is jointly concave in \(c\) and \(v\) if and only if \(\gamma\psi\leq1\).

The decomposition (2.9) is widely used for (time-separable) power utilities; cf. e.g. [35].

When \(h\) is bounded from below, for example, if both \(r\) and \(\mu'\Sigma^{-1} \mu\) are bounded, (2.15) implies that \(Y\) is bounded from below as well. Then \((\pi, c)\) is permissible if \(c\in\mathcal{C}_{a}\) and \((\mathcal{W}^{\pi, c})^{1-\gamma}\) is of class \(D\) on \([0,T]\). This is exactly the definition of permissibility used in [11] for the time-separable utilities with \(\gamma>1\).

Since \(\psi>1\), (3.1) yields \(\frac{b \lambda' \rho}{a} + \frac{1}{2} \psi\lambda'(\psi1_{n\times n} - (\psi-1) \rho\rho') \lambda \leq\frac{b \lambda' \rho}{a}+ \frac{1}{2} \psi\lambda' \lambda\leq0\). Hence the left-hand side of the inequality in Proposition 3.2(ii) is negative.

The proof is the same as in footnote 7.

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Titel: Consumption–investment optimization with Epstein–Zin utility in incomplete markets
verfasst von: Hao Xing
Publikationsdatum: 12.04.2016
Verlag: Springer Berlin Heidelberg
Erschienen in: Finance and Stochastics / Ausgabe 1/2017
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-016-0297-z

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