Stochastic Processes and Calculus

Academic year
Subject Area
Language of Instruction
Mode of Delivery
ECTS Credits
2nd Cycle Studies - Mestrado

Recommended Prerequisites

Basic knowledge in Probability; experience in using scientific software.

Teaching Methods

Based on printed notes, classes are expository and include examples for applying the acquired knowledge. A complete set of training exercises are provided to students. Their resolution is a crucial complement to the course and helps them to prepare for the written tests. During the semester, students develop a computational assignment involving the simulation of trajectories of some stochastic models studied in classes. 

Weekly, tutorial time is offered in order to help students to overcome their learning difficulties. 

Learning Outcomes

Provide the theoretical foundations and the essential mathematical tools for the study of stochastic processes and to illustrate their application in describing and analyzing time-dependent random phenomena.  We also present the main classes of stochastic processes that are widely employed in several fields of science, technology, and specially finance. Another goal of this unit is to introduce stochastic calculus. Here, some basic theory and techniques of stochastic integration and stochastic differential equations are covered, at an elementary level.

This course allows developing the following skills: ability to calculate; using computational tools; knowledge of mathematical results; ability to generalize and abstract; formulating and solving problems; design and use of mathematical models for real situations. On a personal level, it allows to develop individual initiative, teamwork and independent learning.

Work Placement(s)



Review of conditional distribution and conditional expectation.

Introduction to stochastic processes – motivation and definition; Gaussian processes, strong and weak stationary processes, stationary and independent increment processes, Markov processes and martingales; classical examples: random walk, Poisson processes, Wiener process and geometric Brownian motion.

Discrete time Markov chains – transition probabilities; Chapman-Kolmogorov equations; classification of states; absorption; long-run behavior.

Continuous time Markov chains – transition probability functions; infinitesimal generator; finite dimension distributions; Kolmogorov differential equations; explosion; waiting times; embedded chain; long-run behavior; birth and death chains and queuing chains.

Foundations of stochastic calculus –Riemann-Stieltjes’s integral and bounded variation functions; Itô’s integral; Itô processes and stochastic differentials; Black-Scholes and Langevin’s equations; Girsanov theorem.

Assessment Methods

Final assessment
Grading by final examination includes taking an exam (85% of the final grade) and doing a computational assignment (15% of the final grade).: 100.0%

Continuous assessment
Grading during the semester requires taking 3 small tests, a mid-term exam (85% of the final grade) and doing a computational assignment (done individually or by a team of 2 students and representing 15% of the final grade). : 100.0%


R. Durrett, Essentials of Stochastic Processes, Springer-Verlag, 1999.

D. Foata, A. Fuchs, Processus Stochastiques, segunda edição, Dunod, 2004.

G.R. Grimmett, D.R. Stirzaker, Probability and Random Processes, terceira edição, Clarendon Press, Oxford Science Publications, 2001.

S. Karlin, H.M. Taylor, A First Course in Stochastic Processes, Academic Press, 1975.

S. Karlin, H.M. Taylor, An Introduction to Stochastic Modeling, terceira edição, Academic Press, 1998.

H. Kuo, Introduction to Stochastic Integration, Springer-Verlag, 2006.

T. Mikosch, Elementary Stochastic Calculus, Advanced Series on Statistical Science and Applied Probability, Vol.6, World Scientific, 2006.

D. Muller, Processos Estocásticos e Aplicações, Almedina, 2007.

R. Serfozo, Basics of Applied Stochastic Processes, Springer-Verlag, 2009.