# Itō diffusion

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In mathematics — specifically, in stochastic analysis — an **Itō 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. Itō diffusions are named after the Japanese mathematician Kiyoshi Itō.

## Contents

## Overview

A (**time-homogeneous**) **Itō 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 *Itō process*, not a diffusion. Itō 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 Itō 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

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### Sample continuity

An Itō diffusion *X* is a sample continuous process, i.e., for almost all realisations *B _{t}*(ω) of the noise,

*X*(ω) is a continuous function of the time parameter,

_{t}*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 (sample) continuous, an Itō diffusion *X* satisfies the stronger requirement to be a Feller-continuous process.

For a point *x* ∈ **R**^{n}, let **P**^{x} denote the law of *X* given initial datum *X*_{0} = *x*, and let **E**^{x} denote expectation with respect to **P**^{x}.

Let *f* : **R**^{n} → **R** be a Borel-measurable function that is bounded below and define, for fixed *t* ≥ 0, *u* : **R**^{n} → **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. (See below.)

## The Markov property

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### The Markov property

An Itō 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 *X _{t}* 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,

It is easy to show that *X* is adapted to Σ_{∗} (i.e. each *X _{t}* is Σ

_{t}-measurable), so the natural filtration

*F*

_{∗}=

*F*

_{∗}

^{X}of (Ω, Σ) generated by

*X*has

*F*⊆ Σ

_{t}_{t}for each

*t*≥ 0.

Let *f* : **R**^{n} → **R** be a bounded, Borel-measurable function. Then, for all *t* and *h* ≥ 0, the conditional expectation conditioned on the σ-algebra Σ_{t} and the expectation of the process "restarted" from *X _{t}* 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 → [0, +∞] 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 **R**^{n}.

As before, let *f* : **R**^{n} → **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

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### Definition

Associated to each Itō 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 Itō diffusion *X* is the operator *A*, which is defined to act on suitable functions *f* : **R**^{n} → **R** by

The set of all functions *f* for which this limit exists at a point *x* is denoted *D _{A}*(

*x*), while

*D*denotes the set of all

_{A}*f*for which the limit exists for all

*x*∈

**R**

^{n}. One can show that any compactly-supported

*C*

^{2}(twice differentiable with continuous second derivative) function

*f*lies in

*D*and that

_{A}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 d*X _{t}* = d

*B*, is given by

_{t}i.e., *A* = Δ/2, where Δ denotes the Laplace operator.

### The Kolmogorov and Fokker–Planck equations

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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* ∈ *C*^{2}(**R**^{n}; **R**) has compact support and *u* : [0, +∞) × **R**^{n} → **R** is defined by

then *u*(*t*, *x*) is differentiable with respect to *t*, *u*(*t*, ·) ∈ *D _{A}* for all

*t*, and

*u*satisfies the following partial differential equation, known as

**Kolmogorov's backward equation**:

The Fokker–Planck equation (also known as *Kolmogorov's forward equation*) is in some sense the "adjointTemplate:Disambiguation needed" to the backward equation, and tells us how the probability density functions of *X _{t}* evolve with time

*t*. Let ρ(

*t*, ·) be the density of

*X*with respect to Lebesgue measure on

_{t}**R**

^{n}, i.e., for any Borel-measurable set

*S*⊆

**R**

^{n},

Let *A*^{∗} denote the Hermitian adjoint of *A* (with respect to the *L*^{2} inner product). Then, given that the initial position *X*_{0} has a prescribed density ρ_{0}, ρ(*t*, *x*) is differentiable with respect to *t*, ρ(*t*, ·) ∈ *D _{A}*

_{*}for all

*t*, and ρ satisfies the following partial differential equation, known as the

**Fokker–Planck equation**:

### The Feynman–Kac formula

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The Feynman–Kac formula is a useful generalization of Kolmogorov's backward equation. Again, *f* is in *C*^{2}(**R**^{n}; **R**) and has compact support, and *q* : **R**^{n} → **R** is taken to be a continuous function that is bounded below. Define a function *v* : [0, +∞) × **R**^{n} → **R** by

The **Feynman–Kac formula** states that *v* satisfies the partial differential equation

Moreover, if *w* : [0, +∞) × **R**^{n} → **R** is *C*^{1} in time, *C*^{2} in space, bounded on *K* × **R**^{n} 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*(*x*) = 0 for all *x* ∈ **R**^{n}.

## The characteristic operator

### Definition

The characteristic operator of an Itō 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 Itō diffusion *X* is defined by

where the sets *U* form a sequence of open sets *U _{k}* 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* ∈ **R**^{n} and all sequences {*U _{k}*}. If

**E**

^{x}[τ

_{U}] = +∞ 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 *C*^{2} functions *f*, in which case

### Application: Brownian motion on a Riemannian manifold

Above, the generator (and hence characteristic operator) of Brownian motion on **R**^{n} was calculated to be ½Δ, where Δ denotes the Laplace operator. The characteristic operator is useful in defining Brownian motion on an *m*-dimensional Riemannian manifold (*M*, *g*): a **Brownian motion on** *M* is defined to be a diffusion on *M* whose characteristic operator in local coordinates *x _{i}*, 1 ≤

*i*≤

*m*, is given by ½Δ

_{LB}, where Δ

_{LB}is the Laplace-Beltrami operator given in local coordinates by

where [*g ^{ij}*] = [

*g*]

_{ij}^{−1}in the sense of the inverse of a square matrix.

## The resolvent operator

In general, the generator *A* of an Itō diffusion *X* is not a bounded operator. However, if a positive multiple of the identity operator **I** is subtracted from *A* then the resulting operator is invertible. The inverse of this operator can be expressed in terms of *X* itself using the resolvent operator.

For α > 0, the **resolvent operator** *R*_{α}, acting on bounded, continuous functions *g* : **R**^{n} → **R**, is defined by

It can be shown, using the Feller continuity of the diffusion *X*, that *R*_{α}*g* is itself a bounded, continuous function. Also, *R*_{α} and α**I** − *A* are mutually inverse operators:

- if
*f*:**R**^{n}→**R**is*C*^{2}with compact support, then, for all α > 0,

- if
*g*:**R**^{n}→**R**is bounded and continuous, then*R*_{α}*g*lies in*D*and, for all α > 0,_{A}

## Invariant measures

Sometimes it is necessary to find an invariant measure for an Itō diffusion *X*, i.e. a measure on **R**^{n} that does not change under the "flow" of *X*: i.e., if *X*_{0} is distributed according to such an invariant measure μ_{∞}, then *X _{t}* is also distributed according to μ

_{∞}for any

*t*≥ 0. The Fokker–Planck equation offers a way to find such a measure, at least if it has a probability density function ρ

_{∞}: if

*X*

_{0}is indeed distributed according to an invariant measure μ

_{∞}with density ρ

_{∞}, then the density ρ(

*t*, ·) of

*X*does not change with

_{t}*t*, so ρ(

*t*, ·) = ρ

_{∞}, and so ρ

_{∞}must solve the (time-independent) partial differential equation

This illustrates one of the connections between stochastic analysis and the study of partial differential equations. Conversely, a given second-order linear partial differential equation of the form Λ*f* = 0 may be hard to solve directly, but if Λ = *A*^{∗} for some Itō diffusion *X*, and an invariant measure for *X* is easy to compute, then that measure's density provides a solution to the partial differential equation.

### Invariant measures for gradient flows

An invariant measure is comparatively easy to compute when the process *X* is a stochastic gradient flow of the form

where β > 0 plays the role of an inverse temperature and Ψ : **R**^{n} → **R** is a scalar potential satisfying suitable smoothness and growth conditions. In this case, the Fokker–Planck equation has a unique stationary solution ρ_{∞} (i.e. *X* has a unique invariant measure μ_{∞} with density ρ_{∞}) and it is given by the Gibbs distribution:

where the partition function *Z* is given by

Moreover, the density ρ_{∞} satisfies a variational principle: it minimizes over all probability densities ρ on **R**^{n} the free energy functional *F* given by

where

plays the role of an energy functional, and

is the negative of the Gibbs-Boltzmann entropy functional. Even when the potential Ψ is not well-behaved enough for the partition function *Z* and the Gibbs measure μ_{∞} to be defined, the free energy *F*[ρ(*t*, ·)] still makes sense for each time *t* ≥ 0, provided that the initial condition has *F*[ρ(0, ·)] < +∞. The free energy functional *F* is, in fact, a Lyapunov function for the Fokker–Planck equation: *F*[ρ(*t*, ·)] must decrease as *t* increases. Thus, *F* is an *H*-function for the *X*-dynamics.

### Example

Consider the Ornstein-Uhlenbeck process *X* on **R**^{n} satisfying the stochastic differential equation

where *m* ∈ **R**^{n} and β, κ > 0 are given constants. In this case, the potential Ψ is given by

and so the invariant measure for *X* is a Gaussian measure with density ρ_{∞} given by

Heuristically, for large *t*, *X _{t}* is approximately normally distributed with mean

*m*and variance (βκ)

^{−1}. The expression for the variance may be interpreted as follows: large values of κ mean that the potential well Ψ has "very steep sides", so

*X*is unlikely to move far from the minimum of Ψ at

_{t}*m*; similarly, large values of β mean that the system is quite "cold" with little noise, so, again,

*X*is unlikely to move far away from

_{t}*m*.

## The martingale property

In general, an Itō diffusion *X* is not a martingale. However, for any *f* ∈ *C*^{2}(**R**^{n}; **R**) with compact support, the process *M* : [0, +∞) × Ω → **R** defined by

where *A* is the generator of *X*, is a martingale with respect to the natural filtration *F*_{∗} of (Ω, Σ) by *X*. The proof is quite simple: it follows from the usual expression of the action of the generator on smooth enough functions *f* and Itō's lemma (the stochastic chain rule) that

Since Itō integrals are martingales with respect to the natural filtration Σ_{∗} of (Ω, Σ) by *B*, for *t* > *s*,

Hence, as required,

since *M _{s}* is

*F*-measurable.

_{s}## Dynkin's formula

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Dynkin's formula, named after Eugene Dynkin, gives the expected value of any suitably smooth statistic of an Itō diffusion *X* (with generator *A*) at a stopping time. Precisely, if τ is a stopping time with **E**^{x}[τ] < +∞, and *f* : **R**^{n} → **R** is *C*^{2} with compact support, then

Dynkin's formula can be used to calculate many useful statistics of stopping times. For example, canonical Brownian motion on the real line starting at 0 exits the interval (−*R*, +*R*) at a random time τ_{R} with expected value

Dynkin's formula provides information about the behaviour of *X* at a fairly general stopping time. For more information on the distribution of *X* at a hitting time, one can study the *harmonic measure* of the process.

## Associated measures

### The harmonic measure

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In many situations, it is sufficient to know when an Itō diffusion *X* will first leave a measurable set *H* ⊆ **R**^{n}. That is, one wishes to study the first exit time

Sometimes, however, one also wishes to know the distribution of the points at which *X* exits the set. For example, canonical Brownian motion *B* on the real line starting at 0 exits the interval (−1, 1) at −1 with probability ½ and at 1 with probability ½, so *B*_{τ(−1, 1)} is uniformly distributed on the set {−1, 1}.

In general, if *G* is compactly embedded within **R**^{n}, then the **harmonic measure** (or **hitting distribution**) of *X* on the boundary ∂*G* of *G* is the measure μ_{G}^{x} defined by

for *x* ∈ *G* and *F* ⊆ ∂*G*.

Returning to the earlier example of Brownian motion, one can show that if *B* is a Brownian motion in **R**^{n} starting at *x* ∈ **R**^{n} and *D* ⊂ **R**^{n} is an open ball centred on *x*, then the harmonic measure of *B* on ∂*D* is invariant under all rotations of *D* about *x* and coincides with the normalized surface measure on ∂*D*.

The harmonic measure satisfies an interesting **mean value property**: if *f* : **R**^{n} → **R** is any bounded, Borel-measurable function and φ is given by

then, for all Borel sets *G* ⊂⊂ *H* and all *x* ∈ *G*,

The mean value property is very useful in the solution of partial differential equations using stochastic processes.

### The Green measure and Green formula

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Let *A* be a partial differential operator on a domain *D* ⊆ **R**^{n} and let *X* be an Itō diffusion with *A* as its generator. Intuitively, the Green measure of a Borel set *H* is the expected length of time that *X* stays in *H* before it leaves the domain *D*. That is, the **Green measure** of *X* with respect to *D* at *x*, denoted *G*(*x*, ·), is defined for Borel sets *H* ⊆ **R**^{n} by

or for bounded, continuous functions *f* : *D* → **R** by

The name "Green measure" comes from the fact that if *X* is Brownian motion, then

where *G*(*x*, *y*) is Green's function for the operator ½Δ on the domain *D*.

Suppose that **E**^{x}[τ_{D}] < +∞ for all *x* ∈ *D*. Then the **Green formula** holds for all *f* ∈ *C*^{2}(**R**^{n}; **R**) with compact support:

In particular, if the support of *f* is compactly embedded in *D*,

## See also

## References

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