Itō diffusion
In mathematics — specifically, in — an Itô diffusion is a solution to a specific type of . 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.
A (time-homogeneous) Itô diffusion in n-dimensional Euclidean space Rn is a X : [0, +∞) × Ω → Rn defined on a probability space (Ω, Σ, P) and satisfying a stochastic differential equation of the form
where B is an m-dimensional Brownian motion and b : Rn → Rn and σ : Rn → Rn×m satisfy the usual Lipschitz continuity condition
for some constant C and all x, y ∈ Rn; this condition ensures the existence of a unique X to the stochastic differential equation given above. The vector field b is known as the coefficient of X; the matrix field σ is known as the diffusion coefficient of X. It is important to note that b and σ do not depend upon time; if they were to depend upon time, X would be referred to only as an Itô process, not a diffusion. Itô diffusions have a number of nice properties, which include
In particular, an Itô diffusion is a continuous, strongly Markovian process such that the domain of its characteristic operator includes all twice-continuously differentiable functions, so it is a diffusion in the sense defined by Dynkin (1965).
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