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Financial Mathematics by Roman N. Makarov, Giuseppe Campolieti

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Chapter 10

One-Dimensional Brownian Motion and Related Processes

Brownian motion is a keystone in the foundation of mathematical finance. Brownian motion is a continuous-time stochastic process but it can be constructed as a limiting case of symmetric random walks. Recall that a similar approach was used to obtain the log-normal price model as a limiting case of binomial models. Since the probability distribution of Brownian motion is normal, it is reasonable to begin with reviewing some of the important basic facts about the multivariate normal distribution.

10.1 Multivariate Normal Distributions

10.1.1 Multivariate Normal Distribution

The n-variate normal distribution is determined by an n × 1 mean vector μ and an n × n positive definite covariance ...

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