On the use of measure-valued strategies in bond markets
type: Item
instance of: scholarly article
title: On the use of measure-valued strategies in bond markets
zbMATH Open document ID: 1051.60059
author: M. Pratelli
author: Marzia De Donno
published in: Finance and Stochastics
publication date: 2004-11-24T00:00:00Z
review text: The authors propose a theory of cylindrical stochastic integration recently developed by \textit{R. Mikulevicius} and \textit{B. L. Rozovskii} [in: Séminaire de probabilités XXXII. Lect. Notes Math. 1686, 137--165 (1998; Zbl 0910.60041); in: Stochastic partial differential equations: six perspectives. Math. Surv. Monogr. 64, 243--325 (1999; Zbl 0938.60047)] as mathematical background to the theory of bond markets. The question is what is portfolio and what is relative strategy in the case when the market contains an infinite number of traded securities. The second question is whether there exists a satisfactory integration theory, which allows, from a mathematical point of view to treat interest rate models like stock market models. Some preliminaries of functional analysis are presented. Then the authors briefly describe the main steps which lead to the construction of the cylindrical stochastic integral, and of a good class of integrands. Only the case useful for financial applications is considered. The result is applied to bond market models. For example, the existence and the form of replicating strategy is established for an asymptotically attainable contingent claim. In a sense, the authors obtain a negative result: Measure-valued strategies are sufficient to describe all possible portfolios only when the covariance spaces have finite dimension.
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 91G10
zbMATH DE Number: 2118689
zbMATH Keywords: bond markets
zbMATH Keywords: term structure of interest
zbMATH Keywords: measure-valued portfolio
zbMATH Keywords: cylindrical stochastic integration
zbMATH Keywords: covariance spaces
zbMATH Keywords: market completeness
reviewed by: Yuliya S. Mishura
MaRDI profile type: Publication
full work available at URL: s00780-003-0102-7
OpenAlex ID: W2026135940
DOI: 10.1007/S00780-003-0102-7
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description: scientific article; zbMATH DE number 2118689