*Conducted in term:*2020Z

*Erasmus code:*14.3

*ISCED code:*0311

*ECTS credits:*7

*Language:*English

*Organized by:*Faculty of Economic Sciences

*Related to study programmes:*

# 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

## Course coordinators

## Mode

## Learning outcomes

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.

## Assessment criteria

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%.

## Bibliography

Mandatory literature:

K. Sydsater, P. Hammond, A. Seierstad, A. Strom, Futher mathematics for economic analysis, Prentice Hall, 2005

Supplementary literature:

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

## Additional information

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: