Itô diffusion

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.

A (time-homogeneous) Itô diffusion in n-dimensional Euclidean space ${\displaystyle {\boldsymbol {\textbf {R}}}^{n}}$ is a process 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 → R^n×m satisfy the usual Lipschitz continuity condition

for some constant C and all x, y ∈ Rⁿ; this condition ensures the existence of a unique strong solution X to the stochastic differential equation given above. The vector field b is known as the drift 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).

An Itô diffusion X is a sample continuous process, i.e., for almost all realisations B_t(ω) of the noise, X_t(ω) is a continuous function of the time parameter, t. More accurately, there is a "continuous version" of X, a continuous process Y so that

This follows from the standard existence and uniqueness theory for strong solutions of stochastic differential equations.

In addition to being (sample) continuous, an Itô diffusion X satisfies the stronger requirement to be a Feller-continuous process.