Brownian Motion

It is a rather easy process to create rather realistic 2 dimensional islands using Brownian motion. To create an island you simply use a looping Brownian Motion function. A looping fuction is one that will eventually come back to its starting point after a finite number of steps. To create the 2D Brownian Motion function you simply replace X(t) with X(x,y). An example of an island created with this process is shown below.[3]

This image clearly shows the looping process of the Brownian function. To create the island you simply fill in the area that is enclosed within the loop created by the function. The result is a very nice looking islnd.

"Fractals and Scaling in Finance"

The present volume, "Fractals and Scaling in Finance", is a characteristically idiosyncratic work. At once a compendium of Mandelbrot’s pioneering work and a sampling of new results, the presentation seems modeled on the brilliant avant-garde movie "Last Year in Marienbad", in which the usual flow of time is suspended, and the plot is gradually revealed by numerous but slightly different repetitions of a few underlying events. As Mandelbrot himself admits in the Preface, the presentation allows the reader unusual freedom of choice in the order in which the book is read. In fact, I enjoyed this work most when I read it in random order, juxtaposing viewpoints and analyses separated in time by 3 decades, and making clear the progression of ideas that Mandelbrot has generated. These include the classification of different forms of randomness, their manifestation in terms of distribution theory, their ability to be represented compactly, the notion of trading time, the importance of discontinuities, the relationship between financial time series and turbulent time series, the pathologies of commonly abused distributions, particularly the log-normal, and a catalogue of the methods used to derive scaling distributions, both honest and fallacious.

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