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.
The expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:
Making similar use of the law of total variance, the variance can be calculated as:
Lastly, using the law of total probability, the moment generating function can be given as follows:
Let N, Y, and D be as above. Let μ be the probability measure according to which D is distributed, i.e.
Let δ0 be the trivial probability distribution putting all of the mass at zero. Then the probability distribution of Y(t) is the measure
where the exponential exp(ν) of a finite measure ν on Borel subsets of the real line is defined by
and
is a convolution of measures, and the series converges weakly.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
This course gives an introduction to probability theory and stochastic calculus in discrete and continuous time. We study fundamental notions and techniques necessary for applications in finance such
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson ('pwɑːsɒn; pwasɔ̃). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume.
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.
In probability theory and related fields, a stochastic (stəˈkæstɪk) or random process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.
Learn the basics of plasma, one of the fundamental states of matter, and the different types of models used to describe it, including fluid and kinetic.
Correct prediction of particle transport by surface waves is crucial in many practical applications such as search and rescue or salvage operations and pollution tracking and clean-up efforts. Recent results by Deike et al. (J. Fluid Mech., vol. 829, 2017, ...
Secondary electron emission is an important process that plays a significant role in several plasma-related applications. As measuring the secondary electron yield experimentally is very challenging, quantitative modelling of this process to obtain reliabl ...
In 1948, Claude Shannon laid the foundations of information theory, which grew out of a study to find the ultimate limits of source compression, and of reliable communication. Since then, information theory has proved itself not only as a quest to find the ...