MATH-431 / 5 credits

Teacher: Dalang Robert

Language: English

## 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

## Required courses

• Bachelor programme of the Mathematics section
• Swiss school programme up to "Maturité"

## 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.

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)

## Prerequisite for

• martingales in financial mathematics
• stochastic control

## In the programs

• Semester: Spring
• Exam form: Written (summer session)
• Subject examined: Theory of stochastic calculus
• Lecture: 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
• Lecture: 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
• Lecture: 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
• Lecture: 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
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional

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