# FE540 Probability Theory for Financial Engineering

Course Catalog Description

# Introduction

The goals of this course is to provide FE and FA students with the necessary probability theory background to ensure a better performance in the rest of the FE/FA programs. In particular the concepts of sigma fields or algebras are not covered in undergraduate probability courses these are fundamental for stochastic processes and for properly define random variables. The students will learn to perform probability reasoning and fundamental probability calculations to help them with the derivations in statistics, time series, and stochastic calculus.

Campus Fall Spring Summer
On Campus X
Web Campus X

Instructors

Professor Email Office
Ionut Florescu ifloresc@stevens.edu Babbio 544

# Course Objectives

The goals of this course is to provide FE and FA students with the necessary probability theory background to ensure a better performance for the remaining classed in the programs. In particular, the concepts of sigma fields or algebras are not covered in undergraduate probability courses, these are fundamental notions for stochastic processes, and for a proper introduction of random variables. We will emphasize the applicability of probability theory. The students will learn to perform probability reasoning and fundamental probability calculations to help them with the derivations in statistics, time series, and stochastic calculus.

At end of this class the students will be able to:

• Interpret different probability statements and apply probability distributions to solve problems
• Differentiate between different distributions and produce the most appropriate approach
• Know how to apply the central limit theorem to infer probability confidence statements
• Analyze joint random variables using gaussian random vectors theory
• Know how to transform probability density into characteristic or moment generating function and how to work with them

Course Resources

# Textbook

[FT] Florescu, Ionut and Tudor, Ciprian A. Handbook of Probability, Wiley, 2014, ISBN 1118593146, 9781118593141. I chose to use this book as the primary textbook for two reasons. Each chapter is supposed to be more or less self contained and it contains many details that I believe are useful. Second, each chapter has a section with exercises split into two, First set of problems have solutions while the second set does not. I will assign exercises that do not have solutions but you should be working through the ones that have solutions for practice. We will be using two other textbooks.

[G] Ghahramani, Saeed Fundamentals of Probability: with Stochastic Processes, Third Edition, Chapman and Hill/CRC, Nov. 2015, ISBN 9781498755016 This is an undergraduate textbook and it is very useful for those of you who did not do a serious probability class in undergraduate. The book explains very well the basic probability distributions and concepts. I will be using exercises from the book and you should use it as a source of material and problems. Finally,

[F] Florescu, Ionut Probability and Stochastic Processes, Wiley, Oct. 2014, ISBN-13: 978-0470624555, ISBN-10: 047062455 This book has more material than the main textbook but it isn't as detailed which is why I am using the handbook as the main text. However, this book has a second part about stochastic processes which is I believe very useful for future. I am referring in particular to Markov chains and Markov processes, Poisson process and the Brownian motion.

The final grade will be determined upon the student's performance in the course. We will have multiple assignments and possibly quizzes throughout the course. Most of the grade will be coming from the in class midterm as well as from the final.
Only use the .pdf format when submitting files online. If specified in class you can turn in handwritten assignment in the traditional way. You should be able to transform any document into a pdf file. You can use Adobe Acrobat - should be free to Stevens students as far as I know (please call the students help desk), or a simple alternative: use a pdf printer driver. I write all my documents in LATEX and that typesetting program produces pdf files. A simple alternative (using any typesetting program): search on google for a driver that would print to a pdf file. Such drivers are generally free. Late assignments will not be accepted under any circumstances without prior notice and permission of the instructor. If outside circumstances are affecting your ability to perform in the course, you must contact your instructor before you fall behind. Generally the grade distribution follows the following percentages.

• 30% Assignment
• 25% Midterm
• 40% Final
• 5% Quizzes, class participation

Lecture Outline

Week 1 Axioms of Probability, Sample Spaces, Examples Combinatorial Analysis, Counting Permutations, Combinations, Binomial Coefficient F-1, FT-1, G-1,2
Week 2 Conditional Probability and Independence, Law of Total Probability, Bayes Theorem, Applications FT-2, F-2, G-3
Week 3 Random Variables: Generalities FT-3
Week 4 Discrete Random variables, examples FT-4, G-4,5
Week 5 Continuous Random variables, Examples FT-5, G-6,7
Week 6 Generating Random variables. Catching up. FT-6, F-3
Week 7 MIDTERM
Week 8 Random vectors, Joint distribution FT-7, F-4
Week 9 Conditional distribution, Conditional expectation G-8,9
Week 10 Moment Generating Function, Characteristic Function FT-8,9,F-6
Week 11 Gaussian Random Vectors, Catch up FT-10
Week 12 Statistical Inference, Limit Theorems FT-11,12, F-7,8
Week 13 Poisson Process and Markov Chain F-10,12 G-12.2,12.4
Week 14 Brownian motion F-15, G-12.5