Math 126 a: Stochastic Processes

From the movement of a pollen grain in water to financial stock markets, from birth and death of plants to brain’s activity, some laws of nature combine with random phenomena, yielding complex, sometimes seemingly unpredictable dynamics. Stochastic processes theory provides a mathematical framework for studying random dynamics. 

The objective of this course is to introduce the key concepts and mathematical tools in stochastic processes. We will focus on Markov Processes, and study a variety of systems ranging from finite-state Markov chains to the solutions of stochastic differential equations, with a particular focus on predicting the probability for the system to be in a given state at a given time, stationary behaviors, absorption probabilities and many other fun properties!

(picture of Paul Lévy working at the sunset of a quiet Fall evening on the theory of stochastic processes was generated by ChatGPT)

Math 36a: Probability

Many natural or societal phenomena are inherently uncertain: we often cannot predict exactly if they willl happen, or exactly what will happen. We can, however, often talk about the likelihood of different outcomes. Probability theory provides the mathematical framework for reasoning about uncertainty and taming random phenomena. Developed initially in the 17th century around predictions of outcomes in games of chance, it matured into a beautifully elegant mathematical theory that is still a very active area of research. It is also an incredibly practical tool that has become indispensable in economics, finance, machine learning, artificial intelligence, biology, and medicine.
This course introduces the mathematical foundations of probability, and illustrates these mathematical concepts through a variety of real-world applications. This class covers axioms of probability, probability spaces and measures, discrete and continuous random variables, conditional probability and independence, and culminates on two fundamental limit theorems: the Law of Large Numbers that predicts how random phenomena can aggregate into predictible outcomes, and the Central Limit Theorem, which describes the fluctuations around those outcomes.

(picture of Ludwig Boltzmann chilling while working on his summer class material was generated by ChatGPT)