Spring 2022 – Math 194 – Stochastic Processes

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Instructor: Khoi Vo, M.S.

Email: kvo020@ucr.edu

Zoom Room: TBD

Meeting time: TBD

Office hours: TBD

Course overview: We will attempt to learn about the key elements in stochastic processes including Brownian Motions, Markov chains, Monte Carlo Method, Ito formula, Stochastic Differential Equation, Fokker-Planck Equations. If time permits, we will look into the applications of Stochastic Differential Equation into biology (example, the noisy system of gen cells) and finance.

Prerequisites: Calculus, basic probability is a plus but is not needed as we will go over them at the beginning of the course.

Grading: Grades will be based on participation and midterm and final presentations. 

Materials:

  1. Lecture notes by Maria Cameron
  2. Lecture note on Stochastic Differential Equations by Evans (primary)
  3. Stochastic Tools in Mathematics and Science by Alexandre J. Chorin and Ole H. Hald
  4. Stochastic Processes and Applications by Grigorious A. Pavliotis
  5. Some additional papers to explore (if time permits)

Plan: Note that we may take more or less time on certain topics to suit your needs. This course is most about you getting more knowledge in your favorite research topic.

Notes: Your participation in this class is very important. It may be helpful to the department in considering continuing this project into a summer undergraduate research project. So, let’s try our best in this learning process together. We appreciate your effort in learning and making this learning opportunity be beneficial to your academia and life journey.

• Week 1:

  1. Course introduction
  2. Basic Probability
  3. Reference: section 2 of Chorin’s book and page 2 to 16 Maria’s note. 
  4. Reference: section 2 of Evan book

• Week 2:

  1. More Probability
  2. Page 17 to 32 Maria’s note
  3. Reference: section 2 of Chorin’s book
  4. Reference: section 2 of Evan book

• Week 3:

  1. Computing with Probability: Monte Carlo Integration
  2. Reference: sections 3 of Chorin’s book

• Week 4:  

  1. Brownian Motion with applications
  2. Reference: section 4 of Chorin’s book

• Week 5:  Midterm presentations

• Week 6:  

  1. Stochastic Differential Equation
  2. Reference: Chapter 5 and 6 of Chorin’s book
  3. Reference: Chapter 1 of Evan book

 Week 7:

  1. More on Stochastic Differential Equation
  2. Foker – Planck Equation
  3. Reference: Chapter 5 and 6 of Chorin’s book
  4. Reference: Chapter 1 of Evan book

• Week 8:

  1. Markov Chain
  2. Reference: Chapter 8 of Chorin’s book
  3. Reference: Chapter 4 of Evan book

• Week 9:

  1. Ito calculus
  2. More on SDE
  3. Reference: Maria’s note page 82-100

• Week 10:

  1. Reading on a paper

• Finals week: Final presentation

Midterm: A presentation

Final: Presentation on student’s favorite topic per approval of the instructor.