Numerical integration of stochastic differential equations
Summary
In this course we will introduce and study numerical integrators for stochastic differential equations. These numerical methods are important for many applications.
Content
Review of stochastic calculus; Brownian motion, Stochastic integral; Ito's formula, stochastic differential equations; generator; Feynman-Kac's formula
Numerical methods for stochastic differential equations; Euler Maruyama scheme; strong and weak convergence; stability; Milstein scheme and other integrators, Multi-level Monte-Carlo methods
Other topics that may be addressed if time permits:
numerical integration of non-Lipschits SDEs; approximation of mean exit time and stopped diffusion; long time integration and approximation of invariant measure; numerical integration of jump diffusion processes
Learning Prerequisites
Recommended courses
Numerical Analysis, Advanced probability, Stochastic processes (or equivalent), Stochastic calculus
Learning Outcomes
By the end of the course, the student must be able to:
- Analyze the convergence and the stability properties of stochastiques numerical methods
- Implement numerical methods for solving stochastic differential equations
- Illustrate models based on stochastic differential equations
- Manipulate Ito calculus to be able to perfom computations with stochastic differential equations
- Choose an appropriate numerical method to solve a stochastic differential equation
Teaching methods
Ex cathedra lecture, exercises in classroom and computer lab
Assessment methods
Mini-project + written exam
Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.
Resources
Ressources en bibliothèque
- Stochastic Differential Equations, Theory and applications / Arnold
- Introduction to Stochastic Integration / Kuo
- Numerical Solution of Stochastic Differential Equations / Kloeden
- Stochastic Numerics for Mathematical Physics / Milstein
- An Introduction to Stochastic Differential Equations / Evans
Notes/Handbook
P.E. Kloeden, E. Platen, "Numerical Solution of Stochastic Differential Equations", Springer, 1999.
G.N. Milstein, M.V. Tretyakov, "Stochastic Numerics for Mathematical Physics", Springer, 2004.
D. Higham, P. Kloeden, "An Introduction to the Numerical Simulation of Stochastic Differential Equations", SIAM 2021
L.C. Evans, "An Introduction to Stochastic Differential Equations", AMS, 2013
H-H. Kuo, "Introduction to Stochastic Integration", Springer, 2005.
L. Arnold, "Stochastic Differential Equations, Theory and applications", John Wiley & Sons, 1974
Moodle Link
In the programs
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Numerical integration of stochastic differential equations
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Numerical integration of stochastic differential equations
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Numerical integration of stochastic differential equations
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Numerical integration of stochastic differential equations
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Numerical integration of stochastic differential equations
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Numerical integration of stochastic differential equations
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Numerical integration of stochastic differential equations
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Numerical integration of stochastic differential equations
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Numerical integration of stochastic differential equations
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional