# 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**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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