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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.
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.
Touzi's most cited paper, Applications of Malliavin Calculus to Monte Carlo methods in finance, co-authored by Eric Fournié, Jean-Michel Lasry, Jérôme Lebuchoux and Pierre-Louis Lions, describes an original probabilistic method to compute option contract Greeks: delta, gamma, theta, and vega.
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 ...
The Courant Institute of Mathematical Sciences (commonly known as Courant or CIMS) is the mathematics research school of New York University (NYU). Founded in 1935, it is named after Richard Courant, one of the founders of the Courant Institute and also a mathematics professor at New York University from 1936 to 1972, and serves as a center for research and advanced training in computer ...
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 ...
Itô pioneered the theory of stochastic integration and stochastic differential equations, now known as Itô calculus. Its basic concept is the Itô integral, and among the most important results is a change of variable formula known as Itô's lemma (also known as the Itô formula).