Course: 2021/2022

Advanced Stochastic Processes

(14472)

Requirements (Subjects that are assumed to be known)

Stochastic Processes

GENERAL COMPETENCES.
CG4 - Identify or create the appropriate model for the specific problem that arises in each business activity (finance, marketing, planning and control of production, etc.). Manipulate computationally and analytically established models, taking advantage of the power of statistical methods, optimization, etc., and perform the analysis of the results obtained.
SPECIFIC COMPETENCES.
CE02 - Model and analyze statistical data, both static and dynamic, using statistical techniques
CE09 - Prepare, construct and validate statistical models that reproduce the fundamental characteristics of the problems under analysis.
CE10 - Interpret the results of a quantitative analysis and draw practical conclusions about the real problem for which the statistical models have been constructed. Write reports and communicate the conclusions with the help of advanced graphic representation techniques.
CE14 - Identify and use financial tools to solve problems such as risk estimation, calculating the cost of capital, valuation of assets and / or derivatives or estimating the movement of the interest rate and / or exchange rates.
TRANSVERSAL COMPETENCES.
CT3 - Be able to organize and plan your work, making the right decisions based on the information available, gathering and interpreting relevant data to make judgments and critical thinking within your area of ¿¿study.

Description of contents: programme

1 - Brownian motion
1.1 Definition and properties
1.2 Derived Processes
1.3 Simulation
2 - Martingales in continuous time
2.1 Definition and properties
2.2 Optional sampling theorem
3 - Stochastic Integration
3.1 Definition and properties
3.2 Lema of Itô
3.3 Girsanov's theorem
3.4 Martingale Representation Theorem
4 - Introduction to differential stochastic equations
4.1 Itô's Stochastic Differential Equations
4.2 Linear Differential Equations
4.3 Digital solutions
5 - Applications of stochastic calculus to Finance
5.1 The Black-Scholes formula
5.2 Risk neutral measures
5.3 Pricing Exotic options
5.4 Pricing American options

Learning activities and methodology

Theory (4 ECTS). Lectures.
Practice (2 ECTS). Problem solving lessons.

Assessment System

- % end-of-term-examination 0
- % of continuous assessment (assigments, laboratory, practicals...) 100

Basic Bibliography

- H. Bühlmann. Mathematical Methods in Risk Theory.. Springer. 1996 (2nd. ed)
- R. Durrett. Essentials of stochastic processes. Springer. 2012 (2nd ed.)
- S. Asmussen and H. Albrecher. Ruin Probabilities. World Scientific. 2010 (2nd. ed.)
- S.M. Ross. Stochastic Processes. John Wiley & Sons, inc.. 1996 (2nd. ed.)

- R. Durrett · Essentials of Stochastic Processes : http://www.math.duke.edu/~rtd/EOSP/EOSP2E.pdf

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The course syllabus may change due academic events or other reasons.