Brownian motion and its applications in financial mathematics

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Title: Brownian motion and its applications in financial mathematics
Author: Vadhavkar, Vidyesh
Abstract: Brownian Motion is one of the most useful tools in the arsenal of stochastic models. This phenomenon, named after the English botanist Robert Brown, was originally observed as the irregular motion exhibited by a small particle which is totally immersed in a liquid or gas. Since then it has been applied effectively in such areas as statistical goodness of fi t, analyzing stock price levels and quantum mechanics. In 1905, Albert Einstein first explained the phenomenon by assuming that such an immersed particle was continually being bombarded by the molecules in the surrounding medium. The mathematical theory of this important process was developed by Norbert Weiner and for this reason it is also called the Weiner process. In this thesis, we will discuss some of the important properties of Brownian Motion before turning our attention to the concept of a martingale, which we can think of as being a probabilistic model for a fair game. We will then discuss the Martingale Stopping theorem, which basically says that the expectation of a stopped martingale is equal to its expectation at time zero. We will close by discussing some nontrivial applications of this important theorem.
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Date: 2011-09-16

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