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May 1999 Completeness of securities market models--an operator point of view
Robert Bättig
Ann. Appl. Probab. 9(2): 529-566 (May 1999). DOI: 10.1214/aoap/1029962754


We propose a notion of market completeness which is invariant under change to an equivalent probability measure. 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 our setup, the claims which can be approximated by attainable ones has 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." From this viewpoint the example of Artzner and Heath is no longer paradoxical since all the dimensions are 1. We also illustrate how one can check for injectivity of $T*$ and hence for completeness in the case of price processes on a Brownian filtration (e.g., Black-Scholes, Heath-Jarrow-Morton) and price processes driven by a multivariate point process.


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Robert Bättig. "Completeness of securities market models--an operator point of view." Ann. Appl. Probab. 9 (2) 529 - 566, May 1999.


Published: May 1999
First available in Project Euclid: 21 August 2002

zbMATH: 0941.91019
MathSciNet: MR1687390
Digital Object Identifier: 10.1214/aoap/1029962754

Primary: 90A90
Secondary: 60H30

Keywords: Black-Scholes model , Completeness of securities markets , equivalent martingale measures , Heath-Jarrow-Morton model , weak* topology

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.9 • No. 2 • May 1999
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