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Concept
Itô diffusion
Summary
In mathematics – specifically, in stochastic analysis – an Itô diffusion is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation used in physics to describe the Brownian motion of a particle subjected to a potential in a viscous fluid. Itô diffusions are named after the Japanese mathematician Kiyosi Itô.
Overview
A (time-homogeneous) Itô diffusion in n-dimensional Euclidean space Rn is a process X : [0, +∞) × Ω → Rn defined on a probability space (Ω, Σ, P) and satisfying a stochastic differential equation of the form
:\mathrm{d} X_{t} = b(X_t) , \mathrm{d} t + \sigma (X_{t}) , \mathrm{d} B_{t},
where B is an m-dimensional Brownian motion and b : Rn → Rn and σ : Rn → Rn×m satisfy the usual Lipschitz continuity condition
:| b(x) - b(y) | + | \sigma (x) - \sigma (y) | \leq C | x - y |
for some con
Official source
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