Abstract
The Itô stochastic integral with respect to a Brownian motion is constructed, and its properties are shown in detail. The Itô formula is proved. Its applications include Lévy’s characterization of a Brownian motion, the Burkhölder-Davis-Gundy inequality, and the martingale representation theorem. Next, local times and the Tanaka formula are discussed. The Girsanov theorem on change of measures is proved in the last section.
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