After following this course, students are expected to
1. be able to explain important properties of the basic stochastic processes covered in the course: renewal processes, martingales, Brownian motion.
2. be able to recognize the situations where such models are applicable and, in simple cases, are able to provide mathematical modelling and analysis.
Many real-life processes are genuinely stochastic. Think about the busy period of a web server, interacting particle systems in physics, formation of a social network, population growth, fluctuations in financial markets, and many more. These real-life phenomena can be modelled and analyzed using stochastic processes. In simple words, a stochastic process is a random function of time. In this course, we study three important types of stochastic processes: renewal processes, martingales, and Brownian motion.
Renewal processes describe real-life processes with renewal or regenerative structure, that is, the process tends to restart and repeat from time to time. Renewal processes are widely used e.g. in inventory management, queueing theory, and reliability theory. Martingales are in fact a mathematical representation of a fair stochastic game. In the course, we study several core properties of martingales, which form the basis for stochastic integration and limit theory and enable a wide range of applications of martingales in many areas including physics, finance, and computer science. Important examples of martingales are normalized branching processes, which are at the heart of the analysis of large networks, epidemic spreading, and population growth. Other important examples of martingales are based on random walks, a key concept in probability theory that forms a connection between martingales and Brownian motion – the third topic covered in the course. Brownian motion is one of the basic models frequently used in physics, operations research, and finance. It may represent diffusion, market fluctuations, or describe a limiting behavior of a queueing system in heavy traffic. We formally define Brownian motion and study its basic properties.