Completeness of securities market models--an operator point of view

Abstract

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.