Brownovo gibanje

Šuker, Marija (2015) Brownovo gibanje. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

In this thesis we have defined and described Brownian motion, its construction, some interesting properties and methods for its simulation. At the beginning we have defined Brownian motion and described it as a Gaussian process. For this description we needed definitions of standard normal random vectors, normal random vectors and Gaussian processes. By using these definitions, we could describe Brownian motion as Gaussian process with continuous sample paths, with mean $m=0$ and covariance function $\gamma_{st} = s \wedge t$. In addition, we have proven another properties of standard normal random vectors. Then, we have described Lévy’s construction of Brownian motion, i.e. we have constructed Brownian motion as a uniform limit of continuous functions, to ensure that it automatically has continuous paths. In the second chapter, we have described and proven various interesting properties of Brownian motion. Some of the simplest properties are scaling invariance property, time inversion, symmetry invariance property and renewal invariance property. We have used the renewal invariance property of Brownian motion in the proof of Markov property of Brownian motion. The fact that Brownian motion has Markov property means that Brownian motion does not have memory or history. Furthermore, we have proven strong Markov property of Brownian motion, which we have used to prove the reflection principle of Brownian motion. The reflection principle is property which states that Brownian motion reflected at some stopping time is still a Brownian motion. At the end of the second chapter we have shown that Brownian motion is also a martingale. In the third chapter, we have examined the regularity of Brownian motion. We have divided the chapter into two parts. In the first part we have examined and proven some continuity properties of Brownian motion by using the Lévy’s modul of continuity. In the second part, we have proven, by using results such as Hewitt-Savage 0-1 Law and Fatou’s lemma, that Brownian motion is, almost surely, nowhere differentiable. In the end, we have shown the simplest ways to simulate Brownian motion. The simulation of Brownian motion is based on the independence and normal distribution of increments of Brownian motion. We have shown several examples and algorithms for simulating uniform and normal distribution, and finally we have simulated Brownian motion by using independence and normal distribution of its increments.