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Course: 2025/2026

Advanced Stochastic Processes

(14472)

Bachelor in Statistics and Business (Study Plan 2018) (Plan: 400 - Estudio: 203)

Coordinating teacher: MEILAN VILA, ANDREA

Department assigned to the subject: Statistics Department

Type: Electives

ECTS Credits: 6.0 ECTS

Course: 4º

Semester:

Requirements (Subjects that are assumed to be known)

Probability I Probability II Stochastic Processes

Objectives

- Introduce the fundamental concepts of stochastic calculus, including martingales, Brownian motion, and stochastic integration. - Understand and apply key theoretical and practical tools for the study of stochastic processes. - Analyze and solve stochastic differential equations, with emphasis on applications and numerical methods.

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

- Lectures: Presentation of concepts, theoretical development, and illustrative examples (2.6 ECTS). - Problem-solving sessions: Application of course content through exercises and practical cases (2.6 ECTS). - Continuous assessment sessions: Tests and midterm exams (0.8 ECTS).

Assessment System

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

Calendar of Continuous assessment

The evaluation in the ordinary call is carried out entirely through continuous assessment, which consists of two midterm exams (each accounting for 50% of the final grade).

The grade in the extraordinary call will be determined by a final exam.

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.)

Electronic Resources *

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

Additional Bibliography

B. Oksendal. Stochastic Differential Equations. Springer. 2003

(*) Access to some electronic resources may be restricted to members of the university community and require validation through Campus Global. If you try to connect from outside of the University you will need to set up a VPN

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