Abstract
Real and complex exponential martingales; martingale characterization of Brownian motion; random time-change of martingales; integral representation of martingales; iterated and multiple integrals; change of measure and Girsanov’s theorem; Cameron-Martin theorem; Wald’s identity and Novikov’s condition
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The fundamental characterization of Brownian motion in Theorem 18.3 was proved by Lévy (1954), who also (1940) noted the conformai invariance up to a time change of complex Brownian motion and stated the polarity of singletons. A rigorous proof of Theorem 18.6 was later provided by Kakutani (1944a-b). Kunita and Watanabe (1967) gave the first modern proof of Lévy’s characterization theorem, based on Itô’s formula and exponential martingales. The history of the latter can be traced back to the seminal Cameron and Martin (1944) paper, the source of Theorem
18.22, and to Wald’s (1946, 1947) work in sequential analysis, where the identity of Lemma 18.24 first appeared in a version for random walks.
The integral representation in Theorem 18.10 is essentially due to Itô (1951c), who noted its connection with multiple stochastic integrals and chaos expansions. A one-dimensional version of Theorem 18.12 appears in Doob (1953). The general time-change Theorem 18.4 was discovered independently by Dambis (1965) and Dubins and Schwarz (1965), and a systematic study of isotropic martingales was initiated by Getoor and Sharpe (1972). The multivariate result in Proposition 18.8 was noted by Knight (1971), and a version of Proposition 18.9 for general exchangeable processes appears in Kallenberg (1989). The skew-product representation in Corollary 18.7 is due to Galmarino (1963),
The Cameron-Martin theorem was gradually extended to more general settings by many authors, including Maruyama (1954, 1955), Girsanov (1960), and van Schuppen and Wong (1974). The martingale criterion of Theorem 18.23 was obtained by Novikov (1972).
The material in this chapter is covered by many texts, including the excellent monographs by Karatzas and Shreve (1991) and Revuz and Yor (1999). A more advanced and amazingly informative text is Jacod (1979).
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© 2002 Springer Science+Business Media New York
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Kallenberg, O. (2002). Continuous Martingales and Brownian Motion. In: Foundations of Modern Probability. Probability and Its Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4015-8_18
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DOI: https://doi.org/10.1007/978-1-4757-4015-8_18
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