Itô calculus

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Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance, in stochastic differential equations, and more recently even in machine learning. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes: Y t = ∫ 0 t H s d X s , {\displaystyle Y_{t}=\int _{0}^{t}H_{s}\,dX_{s},} where H is a locally square-integrable process adapted to the filtration generated by X, which is a Brownian...

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