﻿ Calcul Stochastique

## 80-646 Stochastic Calculus I

The course is based on the study of the main tools of probability theory that are used in finance and financial engineering. Although the applications are related to these domains and many examples will be studied in class, it is mainly about mathematics. The main objective of this course is to make the student comfortable with the mathematical concepts commonly used in financial engineering: sigma-field, martingale, stopping time, Brownian motion, stochastic integral, diffusion processes,and  risk neutral measures. The course is divided into two main blocks:  discrete and continuous models. Each of these blocks is again subdivided in two parts: a more theoretical section where the mathematical concepts are introduced and a second section in which the mathematical tools are used.

## Mathematical background (3 weeks)

Chapter 1. Fundamental set, sigma-field, measurable function, probability measure

Probability space
Exercises 1

Chapter 2. Stochastic processes, filtration, stopping time
Stochastic processes
Exercises 2

Chapter 3. Conditional expectation
Conditionnal expectation
Exercises 3

Chapter 4. Discrete time martingales
Martingales
Exercises 4

## Discrete Time Market Models (3 weeks)

Chapter 5. Introduction to risk neutral measures

Binomial model
Exercises 5

Chapter 6. Replication and risk neutral measures

Discrete time models
Exercises 6

Chapter. Snell Envelope

American contingent claim
Exercises 7

## Stochastic Calculus (3 weeks)

Chapter 8. Convergence of Random Variable Series

Convergence

Chapter 9. Brownian Motion
Browian Motion
Exercises 9
brownien1.xls      brownien2.m     brownien3.m

Chapter 10. Ito's Integral

Stochastic integral
Exercises 10

Int_stoch.xls

Chapter 11. Stochastic Differential Equations and Ito's Lemma

SDE
Exercices 11

## Applications to Financial Engineering (environ 3 séances)

Chapter 12. Girsanov Theorem and Change of Measures

Girsanov

Chapter 13. Applications du théorème de représentation des martingales

Martingale representation theorem and hedging
Exercises 13

Chapter 14 Autre application

Bonds