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Inmathematics – specifically, instochastic analysis – anItô diffusion is a solution to a specific type ofstochastic differential equation. That equation is similar to theLangevin equation used inphysics to describe theBrownian motion of a particle subjected to a potential in aviscous fluid. Itô diffusions are named after theJapanesemathematicianKiyosi Itô.

A (time-homogeneous)Itô diffusion inn-dimensionalEuclidean space${\displaystyle {\mathbf{\text{R}}}^n}$ is aprocessX : [0, +∞) × Ω → Rⁿ defined on aprobability space (Ω, Σ, P) and satisfying a stochastic differential equation of the form

${\displaystyle \mathrm{d}{X}_t=b(X_t)\mathrm{d}t+\sigma (X_t)\mathrm{d}{B}_t,}$

whereB is anm-dimensionalBrownian motion andb : Rⁿ → Rⁿ and σ : Rⁿ → R^n×m satisfy the usualLipschitz continuity condition

${\displaystyle |b(x)-b(y)|+|\sigma (x)-\sigma (y)|\le C|x-y|}$

for some constantC and allx,y ∈Rⁿ; this condition ensures the existence of a uniquestrong solutionX to the stochastic differential equation given above. Thevector fieldb is known as thedrift coefficient ofX; thematrix field σ is known as thediffusion coefficient ofX. It is important to note thatb and σ do not depend upon time; if they were to depend upon time,X would be referred to only as anItô process, not a diffusion. Itô diffusions have a number of nice properties, which include

• sample andFeller continuity;
• theMarkov property;
• thestrong Markov property;
• the existence of aninfinitesimal generator;
• the existence of acharacteristic operator;
• Dynkin's formula.

In particular, an Itô diffusion is a continuous, strongly Markovian process such that the domain of its characteristic operator includes alltwice-continuously differentiable functions, so it is adiffusion in the sense defined by Dynkin (1965).

An Itô diffusionX is asample continuous process, i.e., foralmost all realisationsB_t(ω) of the noise,X_t(ω) is acontinuous function of the time parameter,t. More accurately, there is a "continuous version" ofX, a continuous processY so that

${\displaystyle \mathbf{P}[X_t=Y_t]=1\text{ for all }t.}$

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

In addition to being (sample) continuous, an Itô diffusionX satisfies the stronger requirement to be aFeller-continuous process.

For a pointx ∈ Rⁿ, letP^x denote the law ofX given initial datumX₀ = x, and letE^x denoteexpectation with respect toP^x.

Letf : Rⁿ → R be aBorel-measurable function that isbounded below and define, for fixedt ≥ 0,u : Rⁿ → R by

${\displaystyle u(x)={\mathbf{E}}^x[f(X_t)].}$

• Lower semi-continuity: iff is lower semi-continuous, thenu is lower semi-continuous.
• Feller continuity: iff is bounded and continuous, thenu is continuous.

The behaviour of the functionu above when the timet is varied is addressed by the Kolmogorov backward equation, the Fokker–Planck equation, etc. (See below.)

An Itô diffusionX has the important property of beingMarkovian: the future behaviour ofX, given what has happened up to some timet, is the same as if the process had been started at the positionX_t at time 0. The precise mathematical formulation of this statement requires some additional notation:

Let Σ_∗ denote thenaturalfiltration of (Ω, Σ) generated by the Brownian motionB: fort ≥ 0,

${\displaystyle {\mathrm{\Sigma }}_t={\mathrm{\Sigma }}_t^B=\sigma \left\{B_s^{-1}(A)\subseteq \mathrm{\Omega }:0\le s\le t,A\subseteq {\mathbf{R}}^n\text{ Borel}\right\}.}$

It is easy to show thatX isadapted to Σ_∗ (i.e. eachX_t is Σ_t-measurable), so the natural filtrationF_∗ = F_∗^X of (Ω, Σ) generated byX hasF_t ⊆ Σ_t for eacht ≥ 0.

Letf : Rⁿ → R be a bounded, Borel-measurable function. Then, for allt andh ≥ 0, theconditional expectation conditioned on theσ-algebra Σ_t and the expectation of the process "restarted" fromX_t satisfy theMarkov property:

${\displaystyle {\mathbf{E}}^x[f(X_{t+h})|{\mathrm{\Sigma }}_t](\omega )={\mathbf{E}}^{X_t(\omega )}[f(X_h)].}$

In fact,X is also a Markov process with respect to the filtrationF_∗, as the following shows:

The strong Markov property is a generalization of the Markov property above in whicht is replaced by a suitable random time τ : Ω → [0, +∞] known as astopping time. So, for example, rather than "restarting" the processX at timet = 1, one could "restart" wheneverX first reaches some specified pointp ofRⁿ.

As before, letf : 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 allh ≥ 0,

${\displaystyle {\mathbf{E}}^x[f(X_{\tau +h})|{\mathrm{\Sigma }}_{\tau }]={\mathbf{E}}^{X_{\tau }}[f(X_h)].}$

Associated to each Itô diffusion, there is a second-orderpartial differential operator known as thegenerator of the diffusion. The generator is very useful in many applications and encodes a great deal of information about the processX. Formally, theinfinitesimal generator of an Itô diffusionX is the operatorA, which is defined to act on suitable functionsf : Rⁿ → R by

${\displaystyle Af(x)=\underset{t\downarrow 0}{lim}\frac{{\mathbf{E}}^x[f(X_t)]-f(x)}{t}.}$

The set of all functionsf for which this limit exists at a pointx is denotedD_A(x), whileD_A denotes the set of allf for which the limit exists for allx ∈ Rⁿ. One can show that anycompactly-supportedC² (twice differentiable with continuous second derivative) functionf lies inD_A and that

${\displaystyle Af(x)=\sum _i{b}_i(x)\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_i}(x)+\frac{1}{2}\sum _{i,j}{\left(\sigma (x)\sigma (x{)}^{\mathrm{\top }}\right)}_{i,j}\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}(x),}$

or, in terms of thegradient andscalar andFrobeniusinner products,

${\displaystyle Af(x)=b(x)\cdot {\mathrm{\nabla }}_{x}f(x)+\frac{1}{2}\left(\sigma (x)\sigma (x{)}^{\mathrm{\top }}\right):{\mathrm{\nabla }}_x{\mathrm{\nabla }}_{x}f(x).}$

The generatorA for standardn-dimensional Brownian motionB, which satisfies the stochastic differential equation dX_t = dB_t, is given by

${\displaystyle Af(x)=\frac{1}{2}\sum _{i,j}{\delta }_{ij}\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}(x)=\frac{1}{2}\sum _i\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}_i^2}(x)}$,

i.e.,A = Δ/2, where Δ denotes theLaplace operator.

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 ofX evolves in time: it must solve a certainpartial differential equation in which timet and the initial positionx are the independent variables. More precisely, iff ∈ C²(Rⁿ; R) has compact support andu : [0, +∞) × Rⁿ → R is defined by

${\displaystyle u(t,x)={\mathbf{E}}^x[f(X_t)],}$

thenu(t, x) is differentiable with respect tot,u(t, ·) ∈ D_A for allt, andu satisfies the followingpartial differential equation, known asKolmogorov's backward equation:

The Fokker–Planck equation (also known asKolmogorov's forward equation) is in some sense the "adjoint" to the backward equation, and tells us how theprobability density functions ofX_t evolve with timet. Let ρ(t, ·) be the density ofX_t with respect toLebesgue measure onRⁿ, i.e., for any Borel-measurable setS ⊆ Rⁿ,

${\displaystyle \mathbf{P}\left[X_t\in S\right]={\int }_S\rho (t,x)\mathrm{d}x.}$

LetA^∗ denote theHermitian adjoint ofA (with respect to theL²inner product). Then, given that the initial positionX₀ has a prescribed density ρ₀, ρ(t, x) is differentiable with respect tot, ρ(t, ·) ∈ D_A* for allt, and ρ satisfies the following partial differential equation, known as theFokker–Planck equation:

The Feynman–Kac formula is a useful generalization of Kolmogorov's backward equation. Again,f is inC²(Rⁿ; R) and has compact support, andq : Rⁿ → R is taken to be acontinuous function that is bounded below. Define a functionv : [0, +∞) × Rⁿ → R by

${\displaystyle v(t,x)={\mathbf{E}}^x\left[\exp \left(-{\int }_0^{t}q(X_s)\mathrm{d}s\right)f(X_t)\right].}$

TheFeynman–Kac formula states thatv satisfies the partial differential equation

Moreover, ifw : [0, +∞) × Rⁿ → R isC¹ in time,C² in space, bounded onK × Rⁿ for all compactK, and satisfies the above partial differential equation, thenw must bev as defined above.

Kolmogorov's backward equation is the special case of the Feynman–Kac formula in whichq(x) = 0 for allx ∈ Rⁿ.

Thecharacteristic operator of an Itô diffusionX 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 theDirichlet problem.

Thecharacteristic operator${\displaystyle \mathcal{A}}$ of an Itô diffusionX is defined by

${\displaystyle \mathcal{A}f(x)=\underset{U\downarrow x}{lim}\frac{{\mathbf{E}}^x\left[f(X_{{\tau }_U})\right]-f(x)}{{\mathbf{E}}^x[{\tau }_U]},}$

where the setsU form a sequence ofopen setsU_k that decrease to the pointx in the sense that

${\displaystyle U_{k+1}\subseteq U_k\text{ and }\bigcap _{k=1}^{\mathrm{\infty }}{U}_k=\{x\},}$

and

${\displaystyle {\tau }_U=inf\{t\ge 0:X_t\notin U\}}$

is the first exit time fromU forX.${\displaystyle D_{\mathcal{A}}}$ denotes the set of allf for which this limit exists for allx ∈ Rⁿ and all sequences {U_k}. IfE^x[τ_U] = +∞ for all open setsU containingx, define

${\displaystyle \mathcal{A}f(x)=0.}$

The characteristic operator and infinitesimal generator are very closely related, and even agree for a large class of functions. One can show that

${\displaystyle D_A\subseteq D_{\mathcal{A}}}$

and that

${\displaystyle Af=\mathcal{A}f\text{ for all }f\in D_A.}$

In particular, the generator and characteristic operator agree for allC² functionsf, in which case

${\displaystyle \mathcal{A}f(x)=\sum _i{b}_i(x)\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_i}(x)+\frac{1}{2}\sum _{i,j}{\left(\sigma (x)\sigma (x{)}^{\mathrm{\top }}\right)}_{i,j}\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}(x).}$

Above, the generator (and hence characteristic operator) of Brownian motion onRⁿ was calculated to be⁠1/2⁠Δ, where Δ denotes the Laplace operator. The characteristic operator is useful in defining Brownian motion on anm-dimensionalRiemannian manifold (M, g): aBrownian motion onM is defined to be a diffusion onM whose characteristic operator${\displaystyle \mathcal{A}}$ in local coordinatesx_i, 1 ≤ i ≤ m, is given by⁠1/2⁠Δ_LB, where Δ_LB is theLaplace-Beltrami operator given in local coordinates by

${\displaystyle {\mathrm{\Delta }}_{\mathrm{L}\mathrm{B}}=\frac{1}{\sqrt{det(g)}}\sum _{i=1}^m\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\left(\sqrt{det(g)}\sum _{j=1}^m{g}^{ij}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\right),}$

where [g^ij] = [g_ij]⁻¹ in the sense ofthe inverse of a square matrix.

In general, the generatorA of an Itô diffusionX is not abounded operator. However, if a positive multiple of the identity operatorI is subtracted fromA then the resulting operator is invertible. The inverse of this operator can be expressed in terms ofX itself using theresolvent operator.

For α > 0, theresolvent operatorR_α, acting on bounded, continuous functionsg : Rⁿ → R, is defined by

${\displaystyle R_{\alpha }g(x)={\mathbf{E}}^x\left[{\int }_0^{\mathrm{\infty }}{e}^{-\alpha t}g(X_t)\mathrm{d}t\right].}$

It can be shown, using the Feller continuity of the diffusionX, thatR_αg is itself a bounded, continuous function. Also,R_α and αI − A are mutually inverse operators:

• iff : Rⁿ → R isC² with compact support, then, for all α > 0,

${\displaystyle R_{\alpha }(\alpha \mathbf{I}-A)f=f;}$

• ifg : Rⁿ → R is bounded and continuous, thenR_αg lies inD_A and, for all α > 0,

${\displaystyle (\alpha \mathbf{I}-A)R_{\alpha }g=g.}$

Sometimes it is necessary to find aninvariant measure for an Itô diffusionX, i.e. a measure onRⁿ that does not change under the "flow" ofX: i.e., ifX₀ is distributed according to such an invariant measure μ_∞, thenX_t is also distributed according to μ_∞ for anyt ≥ 0. The Fokker–Planck equation offers a way to find such a measure, at least if it has a probability density function ρ_∞: ifX₀ is indeed distributed according to an invariant measure μ_∞ with density ρ_∞, then the density ρ(t, ·) ofX_t does not change witht, so ρ(t, ·) = ρ_∞, and so ρ_∞ must solve the (time-independent) partial differential equation

${\displaystyle A^{\ast }{\rho }_{\mathrm{\infty }}(x)=0,x\in {\mathbf{R}}^n.}$

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 Λf = 0 may be hard to solve directly, but if Λ = A^∗ for some Itô diffusionX, and an invariant measure forX is easy to compute, then that measure's density provides a solution to the partial differential equation.

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

${\displaystyle \mathrm{d}{X}_t=-\mathrm{\nabla }\mathrm{\Psi }(X_t)\mathrm{d}t+\sqrt{2{\beta }^{-1}}\mathrm{d}{B}_t,}$

where β > 0 plays the role of aninverse temperature and Ψ : Rⁿ → 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 theGibbs distribution:

${\displaystyle {\rho }_{\mathrm{\infty }}(x)=Z^{-1}\exp (-\beta \mathrm{\Psi }(x)),}$

where thepartition functionZ is given by

${\displaystyle Z={\int }_{{\mathbf{R}}^n}\exp (-\beta \mathrm{\Psi }(x))\mathrm{d}x.}$

Moreover, the density ρ_∞ satisfies avariational principle: it minimizes over all probability densities ρ onRⁿ thefree energy functionalF given by

${\displaystyle F[\rho ]=E[\rho ]+\frac{1}{\beta }S[\rho ],}$

where

${\displaystyle E[\rho ]={\int }_{{\mathbf{R}}^n}\mathrm{\Psi }(x)\rho (x)\mathrm{d}x}$

plays the role of an energy functional, and

${\displaystyle S[\rho ]={\int }_{{\mathbf{R}}^n}\rho (x)\log \rho (x)\mathrm{d}x}$

is the negative of the Gibbs-Boltzmann entropy functional. Even when the potential Ψ is not well-behaved enough for the partition functionZ and the Gibbs measure μ_∞ to be defined, the free energyF[ρ(t, ·)] still makes sense for each timet ≥ 0, provided that the initial condition hasF[ρ(0, ·)] < +∞. The free energy functionalF is, in fact, aLyapunov function for the Fokker–Planck equation:F[ρ(t, ·)] must decrease ast increases. Thus,F is anH-function for theX-dynamics.

Consider theOrnstein-Uhlenbeck processX onRⁿ satisfying the stochastic differential equation

${\displaystyle \mathrm{d}{X}_t=-\kappa (X_t-m)\mathrm{d}t+\sqrt{2{\beta }^{-1}}\mathrm{d}{B}_t,}$

wherem ∈ Rⁿ and β, κ > 0 are given constants. In this case, the potential Ψ is given by

${\displaystyle \mathrm{\Psi }(x)=\frac{1}{2}\kappa |x-m{|}^2,}$

and so the invariant measure forX is aGaussian measure with density ρ_∞ given by

${\displaystyle {\rho }_{\mathrm{\infty }}(x)={\left(\frac{\beta \kappa }{2\pi }\right)}^{\frac{n}{2}}\exp \left(-\frac{\beta \kappa |x-m{|}^2}{2}\right)}$.

Heuristically, for larget,X_t is approximatelynormally distributed with meanm 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", soX_t is unlikely to move far from the minimum of Ψ atm; similarly, large values of β mean that the system is quite "cold" with little noise, so, again,X_t is unlikely to move far away fromm.

In general, an Itô diffusionX is not amartingale. However, for anyf ∈ C²(Rⁿ; R) with compact support, the processM : [0, +∞) × Ω → R defined by

${\displaystyle M_t=f(X_t)-{\int }_0^{t}Af(X_s)\mathrm{d}s,}$

whereA is the generator ofX, is a martingale with respect to the natural filtrationF_∗ of (Ω, Σ) byX. The proof is quite simple: it follows from the usual expression of the action of the generator on smooth enough functionsf andItô's lemma (the stochasticchain rule) that

${\displaystyle f(X_t)=f(x)+{\int }_0^{t}Af(X_s)\mathrm{d}s+{\int }_0^t\mathrm{\nabla }f(X_s{)}^{\mathrm{\top }}\sigma (X_s)\mathrm{d}{B}_s.}$

Since Itô integrals are martingales with respect to the natural filtration Σ_∗ of (Ω, Σ) byB, fort > s,

${\displaystyle {\mathbf{E}}^x[M_t|{\mathrm{\Sigma }}_s]=M_s.}$

Hence, as required,

${\displaystyle {\mathbf{E}}^x[M_t|F_s]={\mathbf{E}}^x\left[{\mathbf{E}}^x[M_t|{\mathrm{\Sigma }}_s]|F_s\right]={\mathbf{E}}^x[M_s|F_s]=M_s,}$

sinceM_s isF_s-measurable.

Dynkin's formula, named afterEugene Dynkin, gives theexpected value of any suitably smooth statistic of an Itô diffusionX (with generatorA) at a stopping time. Precisely, if τ is a stopping time withE^x[τ] < +∞, andf : Rⁿ → R isC² with compact support, then

${\displaystyle {\mathbf{E}}^x[f(X_{\tau })]=f(x)+{\mathbf{E}}^x\left[{\int }_0^{\tau }Af(X_s)\mathrm{d}s\right].}$

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 theinterval (−R, +R) at a random time τ_R with expected value

${\displaystyle {\mathbf{E}}^0[{\tau }_R]=R^2.}$

Dynkin's formula provides information about the behaviour ofX at a fairly general stopping time. For more information on the distribution ofX at ahitting time, one can study theharmonic measure of the process.

In many situations, it is sufficient to know when an Itô diffusionX will first leave ameasurable setH ⊆ Rⁿ. That is, one wishes to study thefirst exit time

${\displaystyle {\tau }_H(\omega )=inf\{t\ge 0|X_t\notin H\}.}$

Sometimes, however, one also wishes to know the distribution of the points at whichX exits the set. For example, canonical Brownian motionB on the real line starting at 0 exits theinterval (−1, 1) at −1 with probability⁠1/2⁠ and at 1 with probability⁠1/2⁠, soB_τ(−1, 1) isuniformly distributed on the set {−1, 1}.

In general, ifG iscompactly embedded withinRⁿ, then theharmonic measure (orhitting distribution) ofX on theboundary ∂G ofG is the measure μ_G^x defined by

${\displaystyle {\mu }_G^x(F)={\mathbf{P}}^x\left[X_{{\tau }_G}\in F\right]}$

forx ∈ G andF ⊆ ∂G.

Returning to the earlier example of Brownian motion, one can show that ifB is a Brownian motion inRⁿ starting atx ∈ Rⁿ andD ⊂ Rⁿ is anopen ball centred onx, then the harmonic measure ofB on ∂D isinvariant under allrotations ofD aboutx and coincides with the normalizedsurface measure on ∂D.

The harmonic measure satisfies an interestingmean value property: iff : Rⁿ → R is any bounded, Borel-measurable function and φ is given by

${\displaystyle \phi (x)={\mathbf{E}}^x\left[f(X_{{\tau }_H})\right],}$

then, for all Borel setsG ⊂⊂ H and allx ∈ G,

${\displaystyle \phi (x)={\int }_{\mathrm{\partial }G}\phi (y)\mathrm{d}{\mu }_G^x(y).}$

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

LetA be a partial differential operator on a domainD ⊆ Rⁿ and letX be an Itô diffusion withA as its generator. Intuitively, the Green measure of a Borel setH is the expected length of time thatX stays inH before it leaves the domainD. That is, theGreen measure ofX with respect toD atx, denotedG(x, ·), is defined for Borel setsH ⊆ Rⁿ by

${\displaystyle G(x,H)={\mathbf{E}}^x\left[{\int }_0^{{\tau }_D}{\chi }_H(X_s)\mathrm{d}s\right],}$

or for bounded, continuous functionsf : D → R by

${\displaystyle {\int }_{D}f(y)G(x,\mathrm{d}y)={\mathbf{E}}^x\left[{\int }_0^{{\tau }_D}f(X_s)\mathrm{d}s\right].}$

The name "Green measure" comes from the fact that ifX is Brownian motion, then

${\displaystyle G(x,H)={\int }_{H}G(x,y)\mathrm{d}y,}$

whereG(x, y) isGreen's function for the operator⁠1/2⁠Δ on the domainD.

Suppose thatE^x[τ_D] < +∞ for allx ∈ D. Then theGreen formula holds for allf ∈ C²(Rⁿ; R) with compact support:

${\displaystyle f(x)={\mathbf{E}}^x\left[f\left(X_{{\tau }_D}\right)\right]-{\int }_{D}Af(y)G(x,\mathrm{d}y).}$

In particular, if the support off iscompactly embedded inD,

${\displaystyle f(x)=-{\int }_{D}Af(y)G(x,\mathrm{d}y).}$

• Diffusion process

• Dynkin, Eugene B.; trans. J. Fabius; V. Greenberg; A. Maitra; G. Majone (1965).Markov processes. Vols. I, II. Die Grundlehren der Mathematischen Wissenschaften, Bände 121. New York: Academic Press Inc.MR 0193671
• Jordan, Richard; Kinderlehrer, David;Otto, Felix (1998). "The variational formulation of the Fokker–Planck equation".SIAM J. Math. Anal.29 (1): 1–17 (electronic).CiteSeerX 10.1.1.6.8815.doi:10.1137/S0036141096303359.S2CID 13890235.MR 1617171
• Øksendal, Bernt K. (2003).Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer.ISBN 3-540-04758-1.MR 2001996 (See Sections 7, 8 and 9)

• Stochastic differential equations

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