A Compactness Principle for Bounded Sequences of Martingales with Applications

Freddy Delbaen · Walter Schachermayer B

68 citations0/0 files storedBirkhäuser Basel eBooks

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Stochastic processes and financial applications Advanced Banach Space Theory Advanced Harmonic Analysis Research

Abstract

For H 1-bounded sequences of martingales, we introduce a technique, related to the Kadeč-Pełczynski-decomposition for L 1 sequences, that allows us to prove compactness theorems. Roughly speaking, a bounded sequence in H 1 can be split into two sequences, one of which is weakly compact, the other forms the singular part. If the martingales are continuous then the singular part tends to zero in the semi-martingale topology. In the general case the singular parts give rise to a process of bounded variation. The technique allows to give a new proof of the Optional Decomposition Theorem in Mathematical Finance.

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References

1999Continuous Martingales and Brownian Motion 2005Stochastic Integration and Differential Equations 2008A General Version of the Fundamental Theorem of Asset Pricing (1994)1994A general version of the fundamental theorem of asset pricing 1987Limit Theorems for Stochastic Processes 1990Stochastic Integration and Differential Equations 1979Calcul Stochastique et Problèmes de Martingales 1998The fundamental theorem of asset pricing for unbounded stochastic processes

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