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The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes. [ 2 ] [ 49 ] The process also has many applications and is the main stochastic process used in stochastic calculus.
In mathematics, the theory of stochastic processes is an important contribution to probability theory, [29] and continues to be an active topic of research for both theory and applications. [30] [31] [32] The word stochastic is used to describe other terms and objects in mathematics.
See also Category:Stochastic processes. Basic affine jump diffusion; Bernoulli process: discrete-time processes with two possible states. Bernoulli schemes: discrete-time processes with N possible states; every stationary process in N outcomes is a Bernoulli scheme, and vice versa. Bessel process; Birth–death process; Branching process ...
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, [1] resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices , [ 2 ] random ...
A basic definition of a discrete-time martingale is a discrete-time stochastic process ... any stochastic process that is both a submartingale and a supermartingale ...
Suppose that , [,] is given, and we wish to compute .Stochastic computing performs this operation using probability instead of arithmetic. Specifically, suppose that there are two random, independent bit streams called stochastic numbers (i.e. Bernoulli processes), where the probability of a 1 in the first stream is , and the probability in the second stream is .
Sample-continuous process; Sazonov's theorem; Schramm–Loewner evolution; Self-similar process; Single-particle trajectory; Spherical contact distribution function; Spitzer's formula; Stationary increments; Stationary process; Statistical fluctuations; Stochastic control; Stochastic differential equation; Stochastic geometry; Stochastic ...
The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces.