November 29, 2011

New: financial mathematics

Filed under: data analysis,mathematics — Peter Saveliev @ 1:01 pm

I started reading about this very recently but so far all makes a lot of sense.

The mathematics is both solid and interesting, especially stochastic calculus. This is a nice break from my recent experiences with what I would call “the mathematics of computer science”: computer vision and image analysis, machine learning, pattern recognition, and my favorite, PageRank.

One issue in this area I have been wondering about is the importance of randomness.

Consider these graphs:

image:price fluctuation.png

They seem similar to random fluctuation of stock prices which are supposed to follow the random walk model:

200px-Pi_stock.svg.png

But, in fact, the curves are made of just five (5!) sine functions:
sin(t)+10sin(.1t)+.1sin(10t)+5sin(.2)+.2sin(5t):

image:5sines.png

It is interesting how everything is connected. A very important person in this field was Kenneth Arrow. He proved Arrow’s Impossibility Theorem, which is about voting, therefore, related to the ranking problem and PageRank. It is also related to the price equilibrium issue. Meanwhile, the existence of equilibria is frequently is addressed via fixed point theory — the topic of my thesis.