 # Stochastics

Stochastics is the newest cluster of the department. We are engaged in research and education in probability theory, stochastic processes and statistics. Our goal is to understand, describe and analyse random phenomena and structures. More generally, we are interested in the mathematics of randomness.

Stochastics is a word describing all of the above and is not synonymous to statistics. We are, in particular, interested in the interactions of Stochastics with other area of mathematics. As an example, how is the fundamental theorem of algebra rigorously proved using Brownian motion? It is incredible that Stochastics is used to conduct research from Pure Mathematics all the way to applications in physics, statistics (Bayesian and classical, non-parametric), operational research, networks, communications, biology and many other areas.

What is extremely fascinating is the way that rigorous theorems can be proved using the language and techniques of probability theory. The so-called probabilistic method of Paul Erdős depends on the first theorem of probability: namely, that for all probability measures P on some set Ω we have that its value at the empty set is 0. Taking the contrapositive of this axiom we have that if A is a subset of Ω  and there is some probability P such that P(A)>0 then A is nonempty. This is used to show existence theorems in mathematics and has had particular success in combinatorics, computer science and number theory.

In studying probability, one is forced to study fascinating pieces of analysis and deal with functions that typically do not appear in classical analysis. Functions that are continuous but nowhere differentiable form the bread and butter of stochastics and the basis of stochastic calculus, a set of tools that leads to optimal and stochastic control theory of enginering systems , to the modeling of the financial market and to macroscopic (thermodynamic) descriptions of complex systems. Stochastics also studies complexity for natural, human-centric reasons: even when a system is not random it is often described by using randomness much better than in any other way.