Theory of stochastic calculus
Summary
Introduction to the mathematical theory of stochastic calculus: construction of stochastic Ito integral, proof of Ito formula, introduction to stochastic differential equations, Girsanov theorem and Feynman-Kac formula, martingale representation theorem.
Content
- Construction of Brownian motion
- Continuous time martingales
- Ito's theory of integration
- Ito's formula with proof
- Existence and uniqueness theorem for solutions of stochastic differential equations
- Girsanov theorem and Feynman-Kac formula
- Martingale representation theorem
Keywords
stochastic calculus, Ito's integral, stochastic differential equations, Girsanov theorem, Feynman-Kac formula, martingale representation
Learning Prerequisites
Required courses
- Bachelor programme of the Mathematics section
- Swiss school programme up to "Maturité"
Recommended courses
Advanced Probability
Important concepts to start the course
Advanced Probability, Probability and analysis course in the Bachelor programme of the Mathematics section
Learning Outcomes
By the end of the course, the student must be able to:
- Demonstrate mastery of the course material
- Demonstrate mastery of the problems related to the exercices sessions
- Demonstrate mastery of the prerequisites
- Demonstrate the capability of using these notions in other contexts
Transversal skills
- Use a work methodology appropriate to the task.
Teaching methods
Ex cathedra lecture and exercises
Expected student activities
Attend lecture regularly, solve the exercises and write down the solutions, study the previous course material before the next course, go over the material before the exam.
Assessment methods
Written exam
In the case of Article 3 paragraph 5 of the Section Regulations, the teacher decides on the form of the examination he communicates to the students concerned.
Resources
Virtual desktop infrastructure (VDI)
No
Bibliography
- J. Michael Steele, Stochastic Calculus and Financial Applications. Springer (2001)
- B. Oksendal, Stochastic Differential Equations (6th edition). Springer (2003)
- J.-F. LeGall, Brownian Motion, Martingales, and Stochastic Calculus. Springer (2016)
Ressources en bibliothèque
- Stochastic Differential Equations / Øksendal
- Stochastic calculus and financial applications / Steele
- Brownian Motion, Martingales, and Stochastic Calculus
Moodle Link
Prerequisite for
- martingales in financial mathematics
- stochastic control
In the programs
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Theory of stochastic calculus
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Theory of stochastic calculus
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Theory of stochastic calculus
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Theory of stochastic calculus
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Theory of stochastic calculus
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional