# Stochastic Calculus

## Master in Financial Engineering, Year 2009-2010

Link to part II of the course

## News feed

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– It will take place on Friday, January 22, 2010, from 12:15 PM to 15:15 PM, in room CO 3.
– You will be allowed four one-sided A4 pages with handwritten formulas only.
and an “answers session” will be organized on Friday, January 15, at 2:15 PM in room INR 113.
NEW: The solusions have now been posted at the bottom of this page, but watch out that the exam is not only a quiz!

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* Here are codes for simulating stochastic processes in matlab.

* Here are the lecture notes. <!– – Lectures notes in chunks: part 1 (lectures 1 to 4), part 2 (lectures 5 to 7), part 3 (lectures 8 to 10)–> <!–

* Please fill the online evaluation form on IS Academia (evaluation period from Friday, Nov 13 until Sunday, Nov 22).

* October 15: room change: we are back to room CE 1 100 for this session.

* October 1: erratum in Homework 3, Exercise 1, corrected.

* Room change: Due to space limitations, the class will take place in room CE 1 105 from September 24 onwards.

* Course starts Thursday, September 17, 2008, at 8:15 AM in room CE 1 100. –>

## General

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* Just for fun: here is a list of paradoxes involving conditional probability:

* Less fun, but still: here are–>

* 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

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Teaching Assistant E-mail Voice Office Office Hours
tba tba tba tba tba
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## 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]. <!–

N. Bingham, R. Kiesel, “Risk-Neutral Valuation : Pricing and Hedging of Financial Derivatives”, Springer 1998.–>

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’?