Probability (Year 2 - Semester 1) Stochastic Processes (Year 3 - Semester 1)

- 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.

1. Martingales 1.1. Definition of martingale 1.2. Basic properties 1.3. Optional stopping theorem 1.4. Martingale convergence 1.5. Continuous-time martingales 2. Brownian motion 2.1. Definition of Brownian motion 2.2. Basic properties 2.3. Simulation of Brownian motion 2.4. Properties of Brownian motion as a martingale 2.5. Donsker's theorem and applications 3. Stochastic integration 3.1. Motivation and definition of the Itô integral 3.2. Basic properties 3.3. Itô's formula 4. Stochastic differential equations 4.1. Examples of stochastic differential equations 4.2. Existence and uniqueness of solutions 4.3. The Black-Scholes formula 4.4. Numerical methods for solving stochastic differential equations 4.5. Simulation of stochastic differential equations

- 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).