2021年数学学院“吉大学子全球胜任力提升计划”研究生系列短课程（11）

报告题目：Introduction to Brownian motion and Stochastic differential equations

报 告 人：Glinyanaya Ekaterina，Institute of Mathematics, National Academy of Sciences of Ukraine

报告地点：Zoom会议 会议 ID：875 2086 3051 Passcode: 1

校内联系人：韩月才 hanyc@jlu.edu.cn

Abstract: This course aims to provide a solid introduction on the stochastic differential equations and the associated ideas of Ito calculus. Since the theory of stochastic differential equations is based on the concept of Brownian motion, the first part of the course will be devoted to this process. Brownian motion plays a central role in the theory of stochastic processes and we will discuss it in detail. We will consider crucial properties of Brownian motion that allow us to define the stochastic integral. The second part of the course is devoted to discussion of solutions to stochastic differential equations and their properties. the course will be provided with a large number of exercises that will help students to better understand the material.

Lecture 1. Brownian motion: definition and basic properties.

In the lecture we give the definition of Brownian motion and discuss the existence of it. Also we obtain some important properties of trajectories such as non-differentiability.

Lecture 2. Brownian motion as Markov process.

In this lecture we recall the definition of Markov process and consider some simple examples. The main aim of the lecture is to show that Brownian motion satisfies Markov and strong Markov property.

Lecture 3. Brownian motion as martingale.

In this lecture we recall the definition of Martingale and consider some simple examples. We will show that Brownian motion is a martingale and derive important properties that are needed for construction of integral with respect to Brownian motion.

Lecture 4. Integral with respect to Brownian motion

The main aim of this lecture is to construct the Ito integral with respect to Brownian motion. Also we will discuss main properties of such integral.

Lecture 5. Ito formula

In this lecture the most important formula of stochastic integration is discussed. We derive the Ito formula and consider examples of its application.

Lecture 6. Local time of Brownian motion.

In the lecture we define the local time of Brownian motion, the amount of time spent at a given level. Tanaka’s formula and basic properties of a Brownian motion also will be discussed in this lecture

Lecture 7. Stochastic differential equations: different types of solutions.  In this lecture we start discussion of stochastic differential equations from different definitions of its solution: strong and weak solution. Also different types of uniqueness of solutions is discussed. We illustrate all this objects with examples.

Lecture 8. Existence and uniqueness of strong solution.

The main aim of this lecture is to prove the theorem about existence and uniqueness of solution. As example we consider application of this theorem to linear stochastic differential equation.

Lecture 9. Markov property of solution to stochastic differential equation.

In this lecture we discuss Kolmogorov-Chepmen equation and prove it for solution to stochastic differential equation. This gives the Markov property of solution to SDE.

Lecture 10. Properties of trajectories of solutions to stochastic differential equation.

In this lecture we consider one-dimensional diffusions and obtain some properties of its trajectories such as recurrence and transience.

报告人简介：Ekaterina Glinyanaya, Doctor of Philosophy (PhD) in mathematics, is a fellow researcher at the Department of the theory of stochastic processes in the Institute of Mathematics, National Academy of Science of Ukraine. Her current scientific interests are stochastic flows with interactions, random measures. She has publications in international scientific journals and numerous talks on conferences.