Titlebook: Introduction to Stochastic Integration; K. L. Chung,R. J. Williams Book 19831st edition Springer Science+Business Media New York 1983 Mart

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Preliminaries,For each interval . in ? = (?∞,∞) let .(.) denote the σ-field of Borel subsets of .. For each . ∈ ?. = [0, ∞), let .. denote .([0, .]) and let . denote .(?.) =∨... — the smallest σ-field containing .. for all . in ?..

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Applications of the Ito Formula,Theorem 6.1. A process M is a Brownian motion in ? if and only if it is a continuous local martingale with quadratic variation [M] such that

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,Local Time and Tanaka’s Formula,In this chapter . denotes a Brownian motion in ?. For each . ∞ ? we shall obtain a decomposition, known as Tanaka’s formula, of the positive submartingale |. — .| as the sum of another Brownian motion . and a continuous increasing process .(·, .). The latter is called the local time of . at ., a fundamental notion invented by P. Lévy (see [19]).

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