# Stochastic Calculus

## Master in Financial Engineering, Year 2009-2010

Link to part II of the course

“-Two pairs, aces high… -Pass. -Ha ha… I told you so: the tide is turning! Three of a kind.
– Hum… chance is not an element whose variations may be analyzed. Randomness is pure… To regulate it would mean that martingales exist…
but they don’t! By the way, talking about chance, do you know that this word comes from the ancient french word “chéance”, meaning “the way dice fall”?”

“In the present economical context, a guy who must do well is the one selling those small panels that one hangs below the big ones.”

## News feed

* Here are codes for simulating stochastic processes in matlab.

* Here are the lecture notes.

## General

* Websites dedicated to probability:

* Here is a brief history of probability.

## Teacher

Name E-mail Voice Office Office Hours
Olivier Lévêque, I&C-LTHI olivier.leveque#epfl.ch 021 693 81 12 INR 132 Tuesday 9:00 AM – 11:00 AM
Thursday 3:00 PM – 5:00 PM

## Schedule

Type Day Hour Room
Lectures Thursday 8:15 AM – 10:00 AM CE 1 105
Exercise Sessions Thursday 10:15 AM – 12:00 PM CE 1 105

## Detailed Program

Date Subject
Thursday, September 17 1. Probability review (1): sigma-fields and random variables
Thursday, September 24 2. Probability review (2): probability measures, distributions, independence, expectation
Thursday, October 1 3. Probability review (3): inequalities, convergence of random variables, limit theorems
Thursday, October 8 4. Probability review (4): conditional expectation
Thursday, October 15 5. Discrete-time processes (1): random walks, filtrations, discrete-time martingales
Thursday, October 22 6. Discrete-time processes (2): stopping times, Doob’s theorems, martingale transforms
Thursday, October 29 Class cancelled
Thursday, November 5 7. Discrete-time processes (3): Markov processes, Gaussian vectors
Thursday, November 12 8. Continuous-time processes (1): Brownian motion, Gaussian processes, Kolmogorov’s theorem
*Tuesday*, November 17
8:15 AM – 12:00 PM
room CM 09
9. Continuous-time processes (2): martingales, Levy’s theorem, Doob’s theorems
Thursday, November 19 10. Continuous-time processes (3): bounded variation processes, quadratic variation
Thursday, November 26 11. Riemann-Stieltjes integral, Ito’s stochastic integral, Fisk-Stratonovic’s stochastic integral
*Tuesday*, December 1
8:15 AM – 12:00 PM
room CM 09
12. Quadratic variation of the Ito integral, Ito-Doeblin’s formula
Thursday, December 10 13. Stochastic differential equations: a first approach through examples
Thursday, December 17 14. Numerical simulation of Brownian motion and stochastic differential equations

## Homeworks (restricted access to the “epfl.ch” and “unil.ch” domains for the solutions)

Problem sets Date Due Solutions
Introductory Quiz Sept 17 Sept 17 Solutions
Homework 1 Sept 17 Sept 24 Solutions 1
Homework 2 Sept 24 Oct 1 Solutions 2
Homework 3 Oct 1 Oct 8 Solutions 3
Homework 4 Oct 8 Oct 15 Solutions 4
Homework 5 Oct 15 Oct 22 Solutions 5
Homework 6 Oct 22 Nov 5 Solutions 6
Homework 7 Nov 5 Nov 12 Solutions 7
Homework 8 Nov 12 Nov 19 Solutions 8
Homework 9 Nov 17 Nov 26 Solutions 9
Homework 10 Nov 19 Nov 26 Solutions 10
Homework 11 Nov 26 Dec 10 Solutions 11
Homework 12 Dec 1 Dec 17 Solutions 12
Quiz Dec 17 Answers sessions on Friday, January 15,
at 2:15 PM in room INR 113
Solutions

## Bibliography

* = introductory, *** = class level, ***** = advanced

### References on probability and measure theory

**** P. Billingsley, “Probability and Measure”, Wiley, 1995.

** N. Bouleau, “Probabilités de l’ingénieur. Variables aléatoires et simulation”, Hermann, 2002.

*** M. Capinski, E. Kopp, “Measure, Integral and Probability”, Springer Verlag, 1999.
(NB: even though this book is tagged as “class level”, we will not cover the material of this book.

** R. Dalang and D. Conus, “Introduction à la théorie des probabilités”, PPUR, 2008.

* R. Durrett, “Essentials of Probability”, Duxbury Press, 1993.

**** R. Durrett, “Probability: Theory and Examples”, Thomson Brooks/Cole, 2004.

** G. Grimmett, D. Stirzaker, “Probability and Random Processes”, Oxford University Press, 2001.

** S. Ross, “A First Course in Probability”, Pearson, 2005.

### References on stochastic calculus

**** A. Bain, “Stochastic Calculus and Stochastic Filtering”

*** R. Bass, Many subjects, including stochastic calculus

***** R. Durrett, “Stochastic Calculus. A Practical Introduction”, CRC Press, 1996.

**** A. Etheridge, “Stochastic Calculus for Finance”

*** F. Klebaner, “Introduction to Stochastic Calculus with Applications”, Imperial College Press, 2005.
(= reference book for the class, available at the polytechnic bookstore “La Fontaine”)

**** J. Goodman, “Stochastic Calculus”

*** H.-H. Kuo, “Introduction to Stochastic Integration”, Springer, 2008.

** S. Lalley, “Course on Mathematical Finance”

** D. Lamberton, B. Lapeyre, “Introduction to Stochastic Calculus Applied to Finance”, Chapman & Hall / CRC Press, 2000.
(this book is translated from french)

*** Th. Mikosch, “Elementary Stochastic Calculus with Finance in View”, World Scientific, 1998.

** B. Oksendal, “Stochastic Differential Equations. An Introduction with Applications”, Springer Verlag, 2003.

*** S. Shreve, “Stochastic Calculus for Finance” (2 volumes), Springer Verlag, 2004.

**** M. Steele, “Stochastic Calculus and Financial Applications”, Springer Verlag, 2001.

*** Lecture notes of a former class on the same topic (in french) [needs revision].

Last updated: January 15, 2010

“Regarding your conversation on randomness [NB: ‘hasard’ in french],
did you know also that ‘ahzir’ is the arabic word for ‘to guess’?