Graduate

Requirements: Real analysis I

G1,2-Graduate Probability I, II

GradP Rigorous proofs for law of large numbers, central limit theorems and martingales - the certain things in uncertain phenomena. Offered Fall 2018, Fall 2019.

GradP2 More limit theorems, Markov chains, Ergodic theory and stationary sequence, Brownian motions, Invariance principles and recent research topics. Offered Spring 2019, Spring 2020.

G3-Large deviations*

The tool to analyze rare events. The concept firstly employed by statisticians to discuss the probability of great loss in insurance theory. Physicists this concept to understand phase traditions. When the temperature varies, the rare event becomes typical.

Keywords: Large deviations, Random matrix theory, Correlated systems, Inequalities

G4-Stochastic Integration*

StochCal Integration with respect to large fluctuations. The universal limit of discrete probability models. SDEs. Brownian-related processes.

G-Continuous-Time Stochastic Processes

We will discuss various types of stochastic processes and mathematical tools for understanding the random phenomenon which depends on time continuously. Providing scientific examples for students to get used to real applications. Systems with continuous parameters are benchmarks for discrete problems. Offered Spring 2021.

Keywords: Stationary, Markov, Point processes

Topics

G- Gaussian processes: Generalization of noise. Offered Spring 2020
Keywords:

G- Point processes: wiki Randomly scatter points in the line/plane/space, how to describe this phenomenon? For any compact set K, we only allow finite points in it. Nowadays, we call it keeping social distance!
Keywords:

Applications

Stochastic financial mathematics

Risk theory

Population dynamics