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MATH 598
Basic Mathematical Tools in Economics and Finance
Course Purpose:
This is a one-semester topic graduate course on the introduction to
the mathematical theories in finance and economics. The course is
designed specificly for those graduate students who wish to learn some basic
mathematical techniques that can be applied in their research and
future career in these areas. Because of the rapid development of computer and
network technologies, more and more mathematical techniques are used
in more and more fields such as physics, chemistry, biology,
economics, finance, engineering, meteorology, rheology, etc.. This
course will cover those materials that
are often not covered in regular undergraduate mathematical courses
but are essential to the areas in finance and economics.
This course has been designed by a group of computational and applied
mathematicians from the department of mathematics with close
collaborations with many professors from different departments at Penn
State University. The syllabus of the course has been extensively
debated and carefully designed to address the special needs of
students with diversely different backgrounds. It is expected this
course will be prerequisite for other basic graduate courses in
computational and applied mathematics offered by the department of
mathematics at Penn State. It is also expected that students who have
taken this course will have a solid foundation to take most
other graduate courses on campus that involve the use of mathematical
and computational techniques.
This course will build a bridge between the basic mathematical
concepts and the applications in economics and finances. Also smooth
the huge gap between the mathematical causes offered on campus and the
demands from the application and research in economics and
finance. This course will give a systematic introduction on all the
subjects that is crucial for the further studying in these
subjects.
Basic Mathematical Tools in Economics and Finance (Spring 2003)
Prerequisite:
Courses equivalent to Math 598B or Econ500; or consent of
the instructor
In this course, we will concentrate on the numerical implication of different
tools, as well as the application to computational economics and computational finance.
- Real and functional analysis:(2 weeks): Metric space; Contraction mapping theorem and
fixed point theorem; applications in numerical simulations to monetary economics models.
(Selected reference: Chapter 7, 9, 10 of Royden, Chapter 3 of Stokey-Lucas-Prescott, Chapter 5 of Gilberg-Trudinger, Chapter 5 and 6 of
Varian))
- Partial differential equations (approximate 2 weeks): basic properties of
solutions; separation of variables; boundary value (eigenvalue) problems; Green's
functions and fundamental solutions.
- Stochastic Calculus (approximate 2 weeks): Probability Spaces and Measure; Random
Variables; Distribution Functions; Special Distributions (Normal, Uniform, Log
normal...); Random Process; Martingale; Brownian Motion; Ito Integral; Ito's Lemma.
- Monte Carlo and Quasi-Monte Carlo Methods (1 weeks): Concepts and Theories; application to
time series simulation and macro-econometric models;
- Introduction to Maple and Matlab (1 weeks) and applications in
linear programming (1 week).
- Stochastic Dynamic Programming (1 weeks): Euler Equation; Principle of
Optimality; Stochastic Euler Equation; Optimal Growth.
- Derivative Pricing (1 weeks): European and American Option; Black-Schoels Model.
- Numerical methods in multi-dimensional derivative pricing (2 week): finite element
approach; Monte Carlo and Quasi-Monte Carlo Methods.
Test and Grades
There will be one midterm exam and one final exam per course. The final
course grade will be determined as follows:
final exam.
Lecture Schedule: TH 11:30 --- 12:40, 113 McAllister Building.
Instructor: Jenny Li,
li@math.psu.edu and
Chun Liu,
liu@math.psu.edu.
Course Texts
1. H. F. Weinberger, A First Course in Partial Differential Equations with
Complex Variables and Transform Methods, Dover.
2. J. Hull, Options, Futures, and Other Derivatives
3. D. Gilberg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer.
4. N. Stokey, R. E. Lucasand E. C. Prescott, Recursive Methods in Economic Dynamics, Harvard.
5. H. R. Varian, Microeconomic Analysis, Second Edition, Norton.
6. Y. Balasko, Foundations of the Theory of General Equilibrium, Academic Press.
7. W. Rudin, Functional Analysis, second Edition, McGraw-Hill.
8. W. Rudin, Real and Complex Analysis, third Edition, McGraw-Hill.
9. H. L. Royden, Real Analysis, third Edition, Macmillan.
Reference Books
Course Steering Committee
© 2003 Center for Computational Mathematics and Applications
Last Updated January 10, 2003 by Jenny
Li
and Chun Liu
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