31 Facts About Stochastic process

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables.

Term random function is used to refer to a stochastic or random process, because a stochastic process can be interpreted as a random element in a function space.

The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.

Stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.

Stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables.

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When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time.

The term stochastic process first appeared in English in a 1934 paper by Joseph Doob.

The first written appearance of the term random process pre-dates stochastic process, which the Oxford English Dictionary gives as a synonym, and was used in an article by Francis Edgeworth published in 1888.

In other words, a Bernoulli Stochastic process is a sequence of iid Bernoulli random variables, where each coin flip is an example of a Bernoulli trial.

Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments.

Almost surely, a sample path of a Wiener Stochastic process is continuous everywhere but nowhere differentiable.

The process has many applications and is the main stochastic process used in stochastic calculus.

The Stochastic process is used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.

Poisson process is a stochastic process that has different forms and definitions.

The number of points of the Stochastic process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter.

For example, a stochastic process can be interpreted or defined as a -valued random variable, where is the space of all the possible functions from the set into the space.

An increment of a stochastic process is the difference between two random variables of the same stochastic process.

Where is a probability measure, the symbol denotes function composition and is the pre-image of the measurable function or, equivalently, the -valued random variable, where is the space of all the possible -valued functions of, so the law of a stochastic process is a probability measure.

Stationarity is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed.

The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line.

One example is when a discrete-time or continuous-time stochastic process is said to be stationary in the wide sense, then the process has a finite second moment for all and the covariance of the two random variables and depends only on the number for all.

Separability is a property of a stochastic process based on its index set in relation to the probability measure.

Markov chain is a type of Markov Stochastic process that has either discrete state space or discrete index set, but the precise definition of a Markov chain varies.

Symmetric random walk and a Wiener Stochastic process are both examples of martingales, respectively, in discrete and continuous time.

Point Stochastic process is a collection of points randomly located on some mathematical space such as the real line, -dimensional Euclidean space, or more abstract spaces.

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The Stochastic process is a sequence of independent Bernoulli trials, which are named after Jackob Bernoulli who used them to study games of chance, including probability problems proposed and studied earlier by Christiaan Huygens.

Bernoulli's work, including the Bernoulli Stochastic process, were published in his book Ars Conjectandi in 1713.

Poisson Stochastic process is named after Simeon Poisson, due to its definition involving the Poisson distribution, but Poisson never studied the Stochastic process.

Stochastic process then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.

Stochastic process introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes.

Furthermore, if a stochastic process is separable, then functionals of an uncountable number of points of the index set are measurable and their probabilities can be studied.