Fiches de cours 2017-2018

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Stochastic calculus I

FIN-408

Enseignant(s) :

Malamud Semyon

Langue:

English

Remarque

For sem. MA1

Summary

This course is an introduction to probability theory and stochastic calculus. It starts with basic notions of probability, characteristic functions and limit theorems. Then, we study stochastic processes and martingales in discrete and continuous time, including Brownian motion and Ito calculus.

Content

1. Probability review (4 weeks) Probability spaces - sigma algebras - random variables - probability measures - independence - Jensen inequality and other basic inequalities for expectations - law of large numbers - central limit theorem - large deviations

 

2. Discrete time processes (4 weeks)Random walks - Markov chains - calculations with stopping times - filtrations - martingales - Gaussian distributions and discrete time Kalman filtering

 

3. Continuous time processes (3 weeks)Brownian motion - continuous filtrations - Gaussian processes - Kolmogorov's theorem - martingales - convergence - optional sampling - Levy's theorem - Doob's theorems - quadratic variation

 

4. Stochastic calculus (3 weeks)Ito's integral - Ito's isometry - Ito's formula - Ito's processes - stochastic differential equations

Keywords

Stochastic calculus, probability

Learning Prerequisites

Important concepts to start the course

Basic analysis, some understanding of probability

Learning Outcomes

By the end of the course, the student must be able to:

Transversal skills

Teaching methods

Ex cathedra classes / exercise sessions

Assessment methods

20% continuous control
40% written mid-term exam
40% written final exam

Supervision

Assistants Yes

Resources

Bibliography

R. Durrett, "Stochastic Calculus. A Practical Introduction", CRC Press, 1996. B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications", Springer Verlag, 2003. S. Shreve, "Stochastic Calculus for Finance" (2 volumes), Springer Verlag, 2004. I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus. Springer Verlag, 1998.

Ressources en bibliothèque

Prerequisite for

Dans les plans d'études

Semaine de référence

 LuMaMeJeVe
8-9     
9-10     
10-11     
11-12     
12-13     
13-14 BS260BS270  
14-15   
15-16     
16-17     
17-18     
18-19     
19-20     
20-21     
21-22     
 
      Cours
      Exercice, TP
      Projet, autre

légende

  • Semestre d'automne
  • Session d'hiver
  • Semestre de printemps
  • Session d'été
  • Cours en français
  • Cours en anglais
  • Cours en allemand