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Mathematics and Financial Economics
Erschienen in: Mathematics and Financial Economics 4/2017

20.06.2017

On the existence of competitive equilibrium in frictionless and incomplete stochastic asset markets

verfasst von: Robert Jarrow

Erschienen in: Mathematics and Financial Economics | Ausgabe 4/2017

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Abstract

Using a standard frictionless, continuous time, and continuous trading stochastic economy with heterogeneous beliefs, the purpose of this paper is to provide sufficient conditions for the existence of competitive equilibrium in an incomplete asset market. A new approach to proving existence is provided, which is readily generalized to markets with frictions, including trading constraints and transaction costs. As a second contribution, this paper also proves the existence of bubble equilibrium in a market without trading constraints. We show that bubbles can exist solely due to heterogeneous beliefs about the evolution of an asset’s market price process.
Vorheriger Artikel Optimal entry to an irreversible investment plan with non convex costs
Nächster Artikel Long-term factorization of affine pricing kernels

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Fußnoten
1
This is a process measurable with respect to the smallest \(\sigma \)-algebra generated by the adapted processes that are right continuous with left limits.
 
2
See Protter [33], p. 163 for the definition of a predictable and integrable process.
 
3
The asymptotic elasticity condition on the utility function can be replaced by the conditions that \(v_{i}(y,S)<\infty \) for all \(y>0\) and \(-\mathbb {\infty }<u_{i}(x,S)\) for all \(x>0\), see Mostovyi [32]. In this case \(\mathcal {M}_{loc}(S)\ne \emptyset \) can be replaced by the set of local martingale deflators being nonempty. The set of local martingale deflators need not be probability density functions with respect to \(\mathbb {P}\).
 
4
Although \(\hat{X}_{T}^{i}(x_{i})\),\(\hat{Y}_{T}^{i}\), and \(y_{i}\) depend on S, for simplicity of notation we omit this dependence. This omission should cause no confusion in the subsequent text.
 
5
When \(\hat{Y_{T}}^{i}\) is a probability density with respect to \(\mathbb {P}\), \(\hat{Y}_{t}^{i}=E\left[ \hat{Y}_{T}^{i}\left| \mathcal {F}_{t}\right. \right] \) is a martingale under \(\mathbb {P}\).
 
6
If we extend the definition of a price bubble to use the local martingale deflator as distinct from the local martingale probability measure, as in Corollary 10 below, then we can remove the probability density condition in the hypothesis of this corollary.
 
7
The existence of an equilibrium in a complete market is a special case of the following results.
 
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Metadaten
Titel
On the existence of competitive equilibrium in frictionless and incomplete stochastic asset markets
verfasst von
Robert Jarrow
Publikationsdatum
20.06.2017
Verlag
Springer Berlin Heidelberg
Erschienen in
Mathematics and Financial Economics / Ausgabe 4/2017
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI
https://doi.org/10.1007/s11579-017-0190-3

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