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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.
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.
He is regarded as one of the fundamental contributors to the theory of diffusion processes with an orientation towards the refinement and further development of Itô’s stochastic calculus. [2] In the year 2007, he became the first Asian to win the Abel Prize. [3] [4]
Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if Q is a measure that is absolutely continuous with respect to P then every P-semimartingale is a Q-semimartingale.
Malliavin introduced Malliavin calculus to provide a stochastic proof that Hörmander's condition implies the existence of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. His calculus enabled Malliavin to prove regularity bounds for the ...
Consider a fixed probability space (,,) and a Hilbert space; denotes expectation with respect to []:= ().Intuitively speaking, the Malliavin derivative of a random variable in () is defined by expanding it in terms of Gaussian random variables that are parametrized by the elements of and differentiating the expansion formally; the Skorokhod integral is the adjoint operation to the Malliavin ...
As with ordinary calculus, integration by parts is an important result in stochastic calculus. The integration by parts formula for the Itô integral differs from the standard result due to the inclusion of a quadratic covariation term. This term comes from the fact that Itô calculus deals with processes with non-zero quadratic variation ...
Kiyosi Itô (伊藤 清, Itō Kiyoshi, Japanese pronunciation: [itoː kiꜜjoɕi], 7 September 1915 – 10 November 2008) was a Japanese mathematician who made fundamental contributions to probability theory, in particular, the theory of stochastic processes.