Lecture

Generation of Markov Processes

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In course

MATH-414: Stochastic simulation

The student who follows this course will get acquainted with computational tools used to analyze systems with uncertainty arising in engineering, physics, chemistry, and economics. Focus will be on sampling methods as Monte Carlo, quasi Monte Carlo, Mark ...

Description

This lecture covers the generation of Markov processes, including discrete time/discrete space Markov chains, continuous time/discrete space Poisson processes, and the definition of Markov chains. It also discusses transition matrices, time homogeneity, stochastic matrices, and the properties of Markov chains. The lecture concludes with examples of Markov chains and transition kernels.

Instructor

Fabio Nobile

Teaching & PhD PhD Students Veronica Montanaro, Matteo Raviola, Francesca Bettinelli, Eliott Van Dieren, Riccardo Saporiti, Giacomo Mossinelli, Noé Stauffer, Fabio Zoccolan Past EPFL PhD Students Diane Sylvie Guignard, Francesco Tesei, Eleonora Musharbash, Matthieu Claude Martin, Davide Pradovera, Sundar Subramaniam Ganesh, Eva Vidlicková, Juan Pablo Madrigal Cianci, Andrea Zanoni, Thomas Simon Spencer Trigo Trindade Past EPFL PhD Students as codirector Michele Pisaroni Courses Advanced analysis II - ve

Official source

Related concepts (31)

Markov property

In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process, which means that its future evolution is independent of its history. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model.

Poisson point process

In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field.

Markov model

In probability theory, a Markov model is a stochastic model used to model pseudo-randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the Markov property). Generally, this assumption enables reasoning and computation with the model that would otherwise be intractable. For this reason, in the fields of predictive modelling and probabilistic forecasting, it is desirable for a given model to exhibit the Markov property.

Compound Poisson process

A compound Poisson process is a continuous-time stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. To be precise, a compound Poisson process, parameterised by a rate and jump size distribution G, is a process given by where, is the counting variable of a Poisson process with rate , and are independent and identically distributed random variables, with distribution function G, which are also independent of When are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process.

Hidden Markov model

A hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process — call it — with unobservable ("hidden") states. As part of the definition, HMM requires that there be an observable process whose outcomes are "influenced" by the outcomes of in a known way.

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