Mathematical Methods in Finance 2400-QFU1MMF
The course is dedicated to advanced undergraduate students of economics.
1. Non-linear programming:
constrained optimization; equality constrains and the Lagrange problem; the constraint qualification; Lagrange multipliers; Kuhn-Tucker multipliers.
2. Differential equations:
Constant coefficient linear differential equation (ODE) systems, fundamental matrix; qualitative solution: phase portrait diagrams; nonlinear systems; fixed points; linearization of dynamic system in the plane.
3. Difference equations:
review of difference equations; linear difference equations; non-linear difference equations and phase diagram; first order difference equations systems.
4. Optimal control:
maximum principle; transversality conditions; transversality conditions in infinite horizon problem; second variations and sufficient conditions.
5. Dynamic programming:
dynamic programming problems; the principle of optimality; the value function; Bellman equation; Euler equations.
6. Stochastic differential equations and partial differential equations:
probability spaces; random variables and stochastic processes; Brownian motion; construction of the Ito integral; the Ito integral; properties of the Ito integral; 1-dimensional Ito processes; 1-dimensional Ito formula; the martingale representation theorem; stochastic differential equations - examples and some solution methods.
Type of course
A student should be able to:
- solve constrained optimization problems,
- solve simple differential and difference equations,
- analyze nonlinear differential and difference equations and systems of equations,
- solve and analyze optimal control problems,
- calculate Ito integrals,
- use the above techniques in economic modeling and finance.
To complete the course, the student has to complete the assignments, pass the midterm exam and pass the final exam. The passing threshold is 50%.
During the midterm and final exam student will have to solve by hand five problems.
Additionally, there will be home assignments. A student will be asked to solve some problems from economics and finance in each homework.
There will be no oral exams.
The final grade will be determined as follows:
Midterm Exam - 30%, Final Exam - 50%, Assignments - 20%.
K. Sydsater, P. Hammond, A. Seierstad, A. Strom, Futher mathematics for economic analysis, Prentice Hall, 2005
1. A. Chiang, Elements of dynamic optimization, McGraw-Hill 1992
2. A. Chiang, Fundamental methods of mathematical economics, McGraw-Hill 1967
3. Z. Brzeźniak, T. Zastawiak, Basic stochastic processes., Springer 2003
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: