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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.
Kuo's book Introduction to Stochastic Integration served as an introductory guide to stochastic integration and the Ito calculus offering information about stochastic processes, stochastic differential equations, concepts of finance, signal processing, and electrical engineering in various fields.
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.
Daniel Wyler Stroock (born March 20, 1940) is an American mathematician, a probabilist.He is regarded and revered as one of the fundamental contributors to Malliavin calculus with Shigeo Kusuoka and the theory of diffusion processes with S. R. Srinivasa Varadhan with an orientation towards the refinement and further development of Itô’s stochastic calculus.
Stochastic mechanics is the framework concerned with the construction of such stochastic processes that generate a probability measure for quantum mechanics. For a Brownian motion , it is known that the statistical fluctuations of a Brownian particle are often induced by the interaction of the particle with a large number of microscopic particles.
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables.
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.
In probability theory, a McKean–Vlasov process is a stochastic process described by a stochastic differential equation where the coefficients of the diffusion depend on the distribution of the solution itself. [1] [2] The equations are a model for Vlasov equation and were first studied by Henry McKean in 1966. [3]