Sample continuity

An Itô diffusion X is a sample continuous process, i.e., for almost all realisations B_t(ω) of the noise, X_t(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

${\displaystyle \mathbf {P} [X_{t}=Y_{t}]=1{\mbox{ for all }}t.}$

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 P^x denote the law of X given initial datum X₀ = x, and let E^x denote expectation with respect to P^x.

Let f : Rⁿ → R be a Borel-measurable function that is bounded below and define, for fixed t ≥ 0, u : Rⁿ → R by

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

• 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

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 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,

${\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\}.}

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_t ⊆ Σ_t 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 Σ_t and the expectation of the process "restarted" from X_t satisfy the Markov property:

${\displaystyle \mathbf {E} ^{x}{\big [}f(X_{t+h}){\big |}\Sigma _{t}{\big ]}(\omega )=\mathbf {E} ^{X_{t}(\omega )}[f(X_{h})].}$

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

${\displaystyle {\begin{aligned}\mathbf {E} ^{x}\left[f(X_{t+h}){\big |}F_{t}\right]&=\mathbf {E} ^{x}\left[\mathbf {E} ^{x}\left[f(X_{t+h}){\big |}\Sigma _{t}\right]{\big |}F_{t}\right]\\&=\mathbf {E} ^{x}\left[\mathbf {E} ^{X_{t}}\left[f(X_{h})\right]{\big |}F_{t}\right]\\&=\mathbf {E} ^{X_{t}}\left[f(X_{h})\right].\end{aligned}}}$

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 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,

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

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

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

The set of all functions f for which this limit exists at a point x is denoted D_A(x), while D_A denotes the set of all f for which the limit exists for all x ∈ Rⁿ. One can show that any compactly-supported C² (twice differentiable with continuous second derivative) function f lies in D_A and that

${\displaystyle Af(x)=\sum _{i}b_{i}(x){\frac {\partial f}{\partial x_{i}}}(x)+{\tfrac {1}{2}}\sum _{i,j}\left(\sigma (x)\sigma (x)^{\top }\right)_{i,j}{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x),}$

or, in terms of the gradient and scalar and Frobenius inner products,

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

An example

The generator A for standard n-dimensional Brownian motion B, which satisfies the stochastic differential equation dX_t = dB_t, is given by

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

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) has compact support and u : [0, +∞) × Rⁿ → R is defined by

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

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:

${\displaystyle {\begin{cases}{\dfrac {\partial u}{\partial t}}(t,x)=Au(t,x),&t>0,x\in \mathbf {R} ^{n};\\u(0,x)=f(x),&x\in \mathbf {R} ^{n}.\end{cases}}}$

The Fokker–Planck equation (also known as Kolmogorov's forward equation) is in some sense the "adjoint" 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_t with respect to Lebesgue measure on Rⁿ, i.e., for any Borel-measurable set S ⊆ Rⁿ,

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

Let A^∗ denote the Hermitian adjoint of A (with respect to the L² inner product). Then, given that the initial position X₀ has a prescribed density ρ₀, 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:

${\displaystyle {\begin{cases}{\dfrac {\partial \rho }{\partial t}}(t,x)=A^{*}\rho (t,x),&t>0,x\in \mathbf {R} ^{n};\\\rho (0,x)=\rho _{0}(x),&x\in \mathbf {R} ^{n}.\end{cases}}}$

The Feynman–Kac formula

The Feynman–Kac formula is a useful generalization of Kolmogorov's backward equation. Again, f is in C²(Rⁿ; R) and has compact support, and q : Rⁿ → R is taken to be a continuous function that is bounded below. Define a function v : [0, +∞) × Rⁿ → R by

${\displaystyle v(t,x)=\mathbf {E} ^{x}\left[\exp \left(-\int _{0}^{t}q(X_{s})\,\mathrm {d} s\right)f(X_{t})\right].}$

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

${\displaystyle {\begin{cases}{\dfrac {\partial v}{\partial t}}(t,x)=Av(t,x)-q(x)v(t,x),&t>0,x\in \mathbf {R} ^{n};\\v(0,x)=f(x),&x\in \mathbf {R} ^{n}.\end{cases}}}$

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(x) = 0 for all x ∈ Rⁿ.

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 ${\displaystyle {\mathcal {A}}}$ of an Itô diffusion X is defined by

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

where the sets U form a sequence of open sets U_k that decrease to the point x in the sense that

${\displaystyle U_{k+1}\subseteq U_{k}{\mbox{ and }}\bigcap _{k=1}^{\infty }U_{k}=\{x\},}$

and

${\displaystyle \tau _{U}=\inf\{t\geq 0\$ :\ X_{t}\not \in U\}}

is the first exit time from U for X. ${\displaystyle D_{\mathcal {A}}}$ denotes the set of all f for which this limit exists for all x ∈ Rⁿ and all sequences {U_k}. If E^x[τ_U] = +∞ for all open sets U containing x, define

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

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

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

and that

${\displaystyle Af={\mathcal {A}}f{\mbox{ for all }}f\in D_{A}.}$

In particular, the generator and characteristic operator agree for all C² functions f, in which case

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

Application: Brownian motion on a Riemannian manifold

The characteristic operator of a Brownian motion is ½ times the Laplace-Beltrami operator. Here it is the Laplace-Beltrami operator on a 2-sphere.

Above, the generator (and hence characteristic operator) of Brownian motion on Rⁿ 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 ${\displaystyle {\mathcal {A}}}$ 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

${\displaystyle \Delta _{\mathrm {LB} }={\frac {1}{\sqrt {\det(g)}}}\sum _{i=1}^{m}{\frac {\partial }{\partial x_{i}}}\left({\sqrt {\det(g)}}\sum _{j=1}^{m}g^{ij}{\frac {\partial }{\partial x_{j}}}\right),}$

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

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

${\displaystyle \mathrm {d} X_{t}=-\nabla \Psi (X_{t})\,\mathrm {d} t+{\sqrt {2\beta ^{-1}}}\,\mathrm {d} B_{t},}$

where β > 0 plays the role of an inverse 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 the Gibbs distribution:

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

where the partition function Z is given by

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

Moreover, the density ρ_∞ satisfies a variational principle: it minimizes over all probability densities ρ on Rⁿ the free energy functional F given by

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

where

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

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 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 Rⁿ satisfying the stochastic differential equation

${\displaystyle \mathrm {d} X_{t}=-\kappa (X_{t}-m)\,\mathrm {d} t+{\sqrt {2\beta ^{-1}}}\,\mathrm {d} B_{t},}$

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

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

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

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

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_t is unlikely to move far from the minimum of Ψ at m; 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 from m.

The harmonic measure

In many situations, it is sufficient to know when an Itô diffusion X will first leave a measurable set H ⊆ Rⁿ. That is, one wishes to study the first exit time

${\displaystyle \tau _{H}(\omega )=\inf\{t\geq 0|X_{t}\not \in H\}.}$

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 Rⁿ, then the harmonic measure (or hitting distribution) of X on the boundary ∂G of G is the measure μ_G^x defined by

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

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 Rⁿ starting at x ∈ Rⁿ and D ⊂ Rⁿ 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 : Rⁿ → R is any bounded, Borel-measurable function and φ is given by

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

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

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

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

The Green measure and Green formula

Let A be a partial differential operator on a domain D ⊆ Rⁿ and let X be an Itô 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 ⊆ 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 functions f : 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 if X is Brownian motion, then

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

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 ∈ 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 of f is compactly embedded in D,

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