Abstract
This chapter starts with a brief history of Brownian motion and then describes its main properties. Ito’s lemma for one dimension and multidimensional Brownian motion are then derived. The Ito product and quotient rules are proved and examples of using them are provided. The Brownian bridge equation is proved and time-transformed and scaled Brownian motion are discussed. The properties of Ornstein–Uhlenbeck processes, such as the mean and the variance, are then derived. The equation for an Ornstein–Uhlenbeck bridge derived and its relationship to the Brownian bridge discussed. Finally Fubini’s theorem, Ito’s isometry and the expectation of stochastic integrals are discussed.
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