On the use of measure-valued strategies in bond markets
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.
scientific article; zbMATH DE number 2118689