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 {\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
d X t = b ( X t ) d t + σ ( X t ) d B t , {\displaystyle \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 ) | + | σ ( x ) − σ ( y ) | ≤ C | x − y | {\displaystyle |b(x)-b(y)|+|\sigma (x)-\sigma (y)|\leq C|x-y|}
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. b and σ do not depend upon time; if they were, 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).
Continuity
Sample continuity An Itô diffusion X is a sample continuous process, i.e., for almost all realisations Bt(ω) of the noise, Xt(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
P [ X t = Y t ] = 1 for all t . {\displaystyle \mathbf {P} [X_{t}=Y_{t}]=1{\mbox{ for all }}t.}
This follows from the standard existence and uniqueness theory for strong solutions of stochastic differential equations.
Feller continuity In addition to being (sample) continuous, an Itô diffusion X satisfies the stronger requirement to be a Feller-continuous process. For a point x ∈ Rn, let Px denote the law of X given initial datum X0 = x, and let Ex denote expectation with respect to Px. Let f : Rn → R be a Borel-measurable function that is bounded below and define, for fixed t ≥ 0, u : Rn → R by
u ( x ) = E x [ f ( X t ) ] . {\displaystyle u(x)=\mathbf {E} ^{x}[f(X_{t})].}
Lower semi-continuity: if f is lower semi-continuous, then u is lower semi-continuous. Feller continuity: if f is bounded and continuous, then u is continuous. The behaviour of the function u above when the time t is varied is addressed by the Kolmogorov backward equation, the Fokker–Planck equation, etc. (See below.)
The Markov property
The Markov property An Itô diffusion X has the important property of being Markovian: the future behaviour of X, given what has happened up to some time t, is the same as if the process had been started at the position Xt at time 0. The precise mathematical formulation of this statement requires some additional notation: Let Σ∗ denote the natural filtration of (Ω, Σ) generated by the Brownian motion B: for t ≥ 0,
Σ t = Σ t B = σ { B s − 1 ( A ) ⊆ Ω : 0 ≤ s ≤ t , A ⊆ R n Borel } . {\displaystyle \Sigma _{t}=\Sigma _{t}^{B}=\sigma \left\{B_{s}^{-1}(A)\subseteq \Omega \ :\ 0\leq s\leq t,A\subseteq \mathbf {R} ^{n}{\mbox{ Borel}}\right\}.}
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