stochastic processes - Definition. Was ist stochastic processes
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Was (wer) ist stochastic processes - definition

MATHEMATICAL OBJECT USUALLY DEFINED AS A COLLECTION OF RANDOM VARIABLES
Random function; Theory of random functions; Stochastic processes; Random process; Stochastic transition function; Heterogeneous process; Stochastic effects; Stochastic Process; Random signal; Random system; Random processes; Stochastic model; Stochastic systems; Homogeneous process; Stochastic models; Kolmogorov extension; Stochastic system; Process (stochastic); Discrete-time stochastic process; Stochastic dynamics; Stochastic deaths; Stochastic processe; Stochastic Processes; Real-valued stochastic process; Version (probability theory)
  • Wiener]] or [[Brownian motion]] process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory.<ref name="doob1953stochasticP46to47"/><ref name="RogersWilliams2000page1"/><ref name="Steele2012page29"/>
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  • Mathematician [[Joseph Doob]] did early work on the theory of stochastic processes, making fundamental contributions, particularly in the theory of martingales.<ref name="Getoor2009"/><ref name="Snell2005"/> His book ''Stochastic Processes'' is considered highly influential in the field of probability theory.<ref name="Bingham2005"/>
  • [[Norbert Wiener]] gave the first mathematical proof of the existence of the Wiener process. This mathematical object had appeared previously in the work of [[Thorvald Thiele]], [[Louis Bachelier]], and [[Albert Einstein]].<ref name="JarrowProtter2004"/>
  • A single computer-simulated '''sample function''' or '''realization''', among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.

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. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner.
Stochastic Models         
Stochastic Models is a peer-reviewed scientific journal that publishes papers on stochastic models. It is published by Taylor & Francis.
List of stochastic processes topics         
WIKIMEDIA LIST ARTICLE
List of stochastic processes; List of stochastic process topics; Stochastic methods
In the mathematics of probability, a stochastic process is a random function. In practical applications, the domain over which the function is defined is a time interval (time series) or a region of space (random field).

Wikipedia

Stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a sequence of random variables; where the index of the sequence have 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. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.

Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.

The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.

Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks, martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes. The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.