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Stochastic Processes
Engineering & Social Sciences Program
Madrid, Spain
Dates: early Sep 2025 - mid Dec 2025
Stochastic Processes
OVERVIEW
CEA CAPA Partner Institution: Universidad Carlos III de Madrid
Location: Madrid, Spain
Primary Subject Area: Mathematics
Instruction in: English
Course Code: 18282
Transcript Source: Partner Institution
Course Details: Level 400
Recommended Semester Credits: 3
Contact Hours: 42
Prerequisites: Probbability (Year 2 - Semester 2)
DESCRIPTION
1. Introduction to Stochastic Processes.
1.1. Basic Definitions and Notations.
1.2. Examples: branching processes and queues.
1.3. Review of Conditional Expectation.
1.4. Review of Characteristic Functions and applications.
2. Discrete time Markov Chains.
2.1. Basic Definitions and Notations.
2.2 Chapman-Kolmogorov Equations and classification of states.
2.3. Asymptotic results.
2.4. First Step Analysis.
2.5. Random Walks and Success Runs.
2.6 The Geo/Geo/1 queue.
3. Renewal Theory and Poisson process.
3.1 Definition and basic notions.
3.2 The Elementary Renewal Theorem.¿
3.3 The Key Renewal Theorem.
3.4 The Delayed Renewal Theorem.
3.5 Compound Poisson Process.
4. Continuous time Markov Chains.
4.1 Definition and basic notions¿
4.2 Chapman-Kolmogorov Equations and Limit Theorems
4.3 Birth and Death Processes (M/M/m queues).
5. Continuous time Markov Processes: Brownian Motion.
5.1 Brownian Motion and Gaussian Processes.
5.2 Variations and Extensions.
5.3 Hitting times.¿
5.4 Relation with Martingales.
1.1. Basic Definitions and Notations.
1.2. Examples: branching processes and queues.
1.3. Review of Conditional Expectation.
1.4. Review of Characteristic Functions and applications.
2. Discrete time Markov Chains.
2.1. Basic Definitions and Notations.
2.2 Chapman-Kolmogorov Equations and classification of states.
2.3. Asymptotic results.
2.4. First Step Analysis.
2.5. Random Walks and Success Runs.
2.6 The Geo/Geo/1 queue.
3. Renewal Theory and Poisson process.
3.1 Definition and basic notions.
3.2 The Elementary Renewal Theorem.¿
3.3 The Key Renewal Theorem.
3.4 The Delayed Renewal Theorem.
3.5 Compound Poisson Process.
4. Continuous time Markov Chains.
4.1 Definition and basic notions¿
4.2 Chapman-Kolmogorov Equations and Limit Theorems
4.3 Birth and Death Processes (M/M/m queues).
5. Continuous time Markov Processes: Brownian Motion.
5.1 Brownian Motion and Gaussian Processes.
5.2 Variations and Extensions.
5.3 Hitting times.¿
5.4 Relation with Martingales.