Course: 2024/2025

Advanced Stochastic Processes

(14472)

Requirements (Subjects that are assumed to be known)

Stochastic Processes

Skills and learning outcomes

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

Calendar of Continuous assessment

Extraordinary call: regulations

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.