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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 R''n'' is a process ''X'' : [0, +∞) × Ω → R''n'' 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'' : R''n'' → R''n'' and σ : R''n'' → R''n''×''m'' satisfy the usual Lipschitz continuity condition : for some constant ''C'' and all ''x'', ''y'' ∈ R''n''; 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 * sample and Feller continuity; * the Markov property; * the strong Markov property; * the existence of an infinitesimal generator; * the existence of a characteristic operator; * Dynkin's formula. 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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Itô diffusion」の詳細全文を読む スポンサード リンク
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