This site contains: mathematics courses and book; covers: image analysis, data analysis, and discrete modelling; provides: image analysis software. Created and run by Peter Saveliev.

Financial mathematics

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:

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

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

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.