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From Wikipedia, the free encyclopedia
In probability theory, Lévy's stochastic area is a stochastic process that describes the enclosed area of a trajectory of a two-dimensional Brownian motion and its chord. The process was introduced by Paul Lévy in 1940,[1] and in 1950[2] he computed the characteristic function and conditional characteristic function.
The process has many unexpected connections to other objects in mathematics such as the soliton solutions of the Korteweg–De Vries equation[3] and the Riemann zeta function.[4] In the Malliavin calculus, the process can be used to construct a process that is smooth in the sense of Malliavin but that has no continuous modification with respect to the Banach norm.[5]
Let be a two-dimensional Brownian motion in then Lévy's stochastic area is the process
where the Itō integral is used.[2]
Define the 1-Form then is the stochastic integral of along the curve
Let , , and then Lévy computed
and
where is the Euclidean norm.[2]: 172–173
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