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Completeness of securities market models – an operator point of view. (English) Zbl 0941.91019

The aim of this paper is to propose a novel approach to a securities market model, based on the notion of market completeness that is invariant under change to an equivalent probability measure.
Section 2 of the paper discusses the new approach to the market completeness, analyzing the second fundamental theorem of asset pricing from an operator theoretic point of view. The simple setting of a securities market model on a finite probability space allows one to point out the fundamental connection between completeness, operators and equivalent martingale measures. The market completeness means that an operator \(T\) acting on stopping time simple trading strategies has dense range in the weak* topology on bounded random variables.
In the exposed approach of Section 3, the space of claims which can be approximated by attainable ones has the codimension equal to the dimension of the kernel of the adjoint operator \(T^*\) acting on signed measures, which in most cases is equal to the dimension of the space of martingale measures.
Section 4 contains the Artzaer-Heath example illustrating the problems that arise when there are an infinite number of discontinuous price processes. In the new market completeness approach, the Artzner-Heath is no longer paradoxical since the space of attainable claims has the codimension one.
The final Sections 5 and 6 show how to check for the injectivity of \(T^*\), hence for the completeness in the case of price processes on a Brownian filtration, and price processes driven by a multivariate point process.

MSC:

91B02 Fundamental topics (basic mathematics, methodology; applicable to economics in general)
60H30 Applications of stochastic analysis (to PDEs, etc.)
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