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.

Overview

A × Ω → Rn defined on a probability space 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 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.

Continuity

Sample continuity

An Itô diffusion X is a sample continuous process, i.e., for almost all realisations Bt of the noise, Xt 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.

Feller continuity

In addition to being 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
  • 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.

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 generated by the Brownian motion B: for t ≥ 0,
It is easy to show that X is adapted to Σ∗, so the natural filtration F∗ = F∗X of generated by X has Ft ⊆ Σt for each t ≥ 0.
Let f : Rn → R be a bounded, Borel-measurable function. Then, for all t and h ≥ 0, the conditional expectation conditioned on the σ-algebra Σt and the expectation of the process "restarted" from Xt satisfy the Markov property:
In fact, X is also a Markov process with respect to the filtration F∗, as the following shows:

The strong Markov property

The strong Markov property is a generalization of the Markov property above in which t is replaced by a suitable random time τ : Ω → known as a stopping time. So, for example, rather than "restarting" the process X at time t = 1, one could "restart" whenever X first reaches some specified point p of Rn.
As before, let f : Rn → R be a bounded, Borel-measurable function. Let τ be a stopping time with respect to the filtration Σ∗ with τ < +∞ almost surely. Then, for all h ≥ 0,

The generator

Definition

Associated to each Itô diffusion, there is a second-order partial differential operator known as the generator of the diffusion. The generator is very useful in many applications and encodes a great deal of information about the process X. Formally, the infinitesimal generator of an Itô diffusion X is the operator A, which is defined to act on suitable functions f : Rn → R by
The set of all functions f for which this limit exists at a point x is denoted DA, while DA denotes the set of all f for which the limit exists for all x ∈ Rn. One can show that any compactly-supported C2 function f lies in DA and that
or, in terms of the gradient and scalar and Frobenius inner products,

An example

The generator A for standard n-dimensional Brownian motion B, which satisfies the stochastic differential equation dXt = dBt, is given by
i.e., A = Δ/2, where Δ denotes the Laplace operator.

The Kolmogorov and Fokker–Planck equations

The generator is used in the formulation of Kolmogorov's backward equation. Intuitively, this equation tells us how the expected value of any suitably smooth statistic of X evolves in time: it must solve a certain partial differential equation in which time t and the initial position x are the independent variables. More precisely, if f ∈ C2 × Rn → R is defined by
then u is differentiable with respect to t, u ∈ DA for all t, and u satisfies the following partial differential equation, known as Kolmogorov's backward equation:
The Fokker–Planck equation is in some sense the "adjoint" to the backward equation, and tells us how the probability density functions of Xt evolve with time t. Let ρ be the density of Xt with respect to Lebesgue measure on Rn, i.e., for any Borel-measurable set S ⊆ Rn,
Let A∗ denote the Hermitian adjoint of A. Then, given that the initial position X0 has a prescribed density ρ0, ρ is differentiable with respect to t, ρ ∈ DA* for all t, and ρ satisfies the following partial differential equation, known as the Fokker–Planck equation:

The Feynman–Kac formula

The Feynman–Kac formula is a useful generalization of Kolmogorov's backward equation. Again, f is in C2 × Rn → R by
The Feynman–Kac formula states that v satisfies the partial differential equation
Moreover, if w : 0, +∞) × Rn → R is C1 in time, C2 in space, bounded on K × Rn for all compact K, and satisfies the above partial differential equation, then w must be v as defined above.
Kolmogorov's backward equation is the [special case of the Feynman–Kac formula in which q = 0 for all x ∈ Rn.

The characteristic operator

Definition

The characteristic operator of an Itô diffusion X is a partial differential operator closely related to the generator, but somewhat more general. It is more suited to certain problems, for example in the solution of the Dirichlet problem.
The characteristic operator of an Itô diffusion X is defined by
where the sets U form a sequence of open sets Uk that decrease to the point x in the sense that
and
is the first exit time from U for X. denotes the set of all f for which this limit exists for all x ∈ Rn and all sequences. If Ex = +∞ for all open sets U containing x, define

Relationship with the generator

The characteristic operator and infinitesimal generator are very closely related, and even agree for a large class of functions. One can show that
and that
In particular, the generator and characteristic operator agree for all C2 functions f, in which case

Application: Brownian motion on a Riemannian manifold

Above, the generator of Brownian motion on Rn was calculated to be Δ, where Δ denotes the Laplace operator. The characteristic operator is useful in defining Brownian motion on an m-dimensional Riemannian manifold : a Brownian motion on M is defined to be a diffusion on M whose characteristic operator in local coordinates xi, 1 ≤ i ≤ m, is given by ΔLB, where ΔLB is the Laplace-Beltrami operator given in local coordinates by
where = −1 in the sense of the inverse of a square matrix.