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Finance and Stochastics
Erschienen in: Finance and Stochastics 1/2017

06.12.2016

Optimal consumption and investment with Epstein–Zin recursive utility

verfasst von: Holger Kraft, Thomas Seiferling, Frank Thomas Seifried

Erschienen in: Finance and Stochastics | Ausgabe 1/2017

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Abstract

We study continuous-time optimal consumption and investment with Epstein–Zin recursive preferences in incomplete markets. We develop a novel approach that rigorously constructs the solution of the associated Hamilton–Jacobi–Bellman equation by a fixed point argument and makes it possible to compute both the indirect utility and, more importantly, optimal strategies. Based on these results, we also establish a fast and accurate method for numerical computations. Our setting is not restricted to affine asset price dynamics; we only require boundedness of the underlying model coefficients.
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Nächster Artikel Consumption–investment optimization with Epstein–Zin utility in incomplete markets

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Fußnoten
1
Our analysis imposes no structural conditions on the underlying model coefficients, but requires them to be bounded; see (A1) and (A2) in Sect. 4 and (A1′) in Sect. 7.
 
2
In particular, [42] covers specifications with (untruncated) affine dynamics as in Kim and Omberg [24] and Heston [23].
 
3
Condition (3.1) holds if and only if one of the conditions (a), (b), (c), and (d) in [26, Proposition 3.2] is satisfied; see also [40, (2)]. We are not aware of rigorous results that ensure (E1) and (E2) for parametrizations not subsumed by (3.1).
 
4
Typically, the pair \((X,Z)\) would be referred to as a solution of the BSDE (6.4). For simplicity of notation, and since \(Z\) is not required for our further analysis, here and in the following, we also refer to \(X\) alone as a solution of (6.4).
 
5
Machine: Intel® Core™ i3-540 Processor (4M Cache, 3.06 GHz), 4 GB RAM.
 
6
Here we slightly abuse notation since \(\langle u\rangle^{q}_{x}\) has only been defined for functions on \([0,T]\times \mathbb{R}^{d}\). Of course, for \(u:\ \mathbb{R}^{d}\to \mathbb{R}\) and \(q\in(0,1)\), we understand that \(\langle u\rangle^{q}_{x} :=\sup_{x,x' \in \mathbb{R}^{d},\ |x-x'| \leq 1} \frac {|u(x) - u(x')|}{|x-x'|^{q}}\).
 
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Metadaten
Titel
Optimal consumption and investment with Epstein–Zin recursive utility
verfasst von
Holger Kraft
Thomas Seiferling
Frank Thomas Seifried
Publikationsdatum
06.12.2016
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/s00780-016-0316-0

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