## Master in Financial Engineering, Year 2009-2010

“-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”?”

© Lewis Trondheim, 1995

“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.”

© Philippe Geluck, 2002

## News feed

* Here are **codes** for simulating stochastic processes in matlab.

* Here are the lecture notes.

## General

* Websites dedicated to probability:

- Introduction to Probability (by C. Grinstead and L. Snell)

Virtual Laboratories in Probability and Statistics

The Probability Web

* Here is a brief history of probability.

## Teacher

Name | 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.

This reference is here if you want to learn more about measure theory).

** 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”

*** L. Evans, “An Introduction to Stochastic Differential Equations”

*** **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’?

© Lewis Trondheim, 1995