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Itô diffusion

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In <a href="Mathematics" title="Mathematics">mathematics</a> – specifically, in <a href="Stochastic_processes" class="mw-redirect" title="Stochastic processes">stochastic analysis</a> – an Itô diffusion is a solution to a specific type of <a href="Stochastic_differential_equation" title="Stochastic differential equation">stochastic differential equation</a>. That equation is similar to the <a href="Langevin_equation" title="Langevin equation">Langevin equation</a> used in <a href="Physics" title="Physics">physics</a> to describe the <a href="Brownian_motion" title="Brownian motion">Brownian motion</a> of a particle subjected to a potential in a <a href="Viscosity" title="Viscosity">viscous</a> fluid. Itô diffusions are named after the <a href="Japan" title="Japan">Japanese</a> <a href="Mathematician" title="Mathematician">mathematician</a> <a href="Kiyosi_It%C3%B4" title="Kiyosi Itô">Kiyosi Itô</a>.

<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>

A (time-homogeneous) Itô diffusion in n-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\textbf {R}}}^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\textbf {R}}}^{n}}</annotation>

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</math><img src="./8acab766a621d7a2491669965ae50dff48ad75b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle {\boldsymbol {\textbf {R}}}^{n}}" loading="lazy"> is a <a href="Stochastic_process" title="Stochastic process">process</a> X : [0, +∞) × Ω → Rn defined on a <a href="Probability_space" title="Probability space">probability space</a> (Ω, Σ, P) and satisfying a stochastic differential equation of the form

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} X_{t}=b(X_{t})\,\mathrm {d} t+\sigma (X_{t})\,\mathrm {d} B_{t},}</annotation>

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</math><img src="./f22f2b2acc63020e549d11302be425a07fbc773c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.864ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} X_{t}=b(X_{t})\,\mathrm {d} t+\sigma (X_{t})\,\mathrm {d} B_{t},}" loading="lazy"></dd></dl>

where B is an m-dimensional <a href="Brownian_motion" title="Brownian motion">Brownian motion</a> and b : Rn → Rn and σ : Rn → Rn×m satisfy the usual <a href="Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuity</a> condition

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<annotation encoding="application/x-tex">{\displaystyle |b(x)-b(y)|+|\sigma (x)-\sigma (y)|\leq C|x-y|}</annotation>

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</math><img src="./c968cd92396bbc2f768b31aefe622c0aadb94dbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.455ex; height:2.843ex;" alt="{\displaystyle |b(x)-b(y)|+|\sigma (x)-\sigma (y)|\leq C|x-y|}" loading="lazy"></dd></dl>

for some constant C and all x, y ∈ Rn; this condition ensures the existence of a unique <a href="Stochastic_differential_equation#Use_in_probability_and_mathematical_finance" title="Stochastic differential equation">strong solution</a> X to the stochastic differential equation given above. The <a href="Vector_field" title="Vector field">vector field</a> b is known as the <a href="Stochastic_drift" title="Stochastic drift">drift</a> coefficient of X; the <a href="Tensor_field" title="Tensor field">matrix field</a> 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 <a href="It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a>, not a diffusion. Itô diffusions have a number of nice properties, which include

<ul><li><a href="Sample_continuous_process" class="mw-redirect" title="Sample continuous process">sample</a> and <a href="Feller-continuous_process" title="Feller-continuous process">Feller continuity</a>;</li>

<li>the <a href="Markov_property" title="Markov property">Markov property</a>;</li>

<li>the <a href="Strong_Markov_property" class="mw-redirect" title="Strong Markov property">strong Markov property</a>;</li>

<li>the existence of an <a href="Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">infinitesimal generator</a>;</li>

<li>the existence of a <a href="#The_characteristic_operator">characteristic operator</a>;</li>

<li><a href="Dynkin's_formula" title="Dynkin's formula">Dynkin's formula</a>.</li></ul>

In particular, an Itô diffusion is a continuous, strongly Markovian process such that the domain of its characteristic operator includes all <a href="Smooth_function" class="mw-redirect" title="Smooth function">twice-continuously differentiable</a> functions, so it is a diffusion in the sense defined by Dynkin (1965).

<div class="mw-heading mw-heading2"><h2 id="Continuity">Continuity</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Continuous_stochastic_process" title="Continuous stochastic process">Continuous stochastic process</a></div>

<div class="mw-heading mw-heading3"><h3 id="Sample_continuity">Sample continuity</h3></div>

An Itô diffusion X is a <a href="Sample_continuous_process" class="mw-redirect" title="Sample continuous process">sample continuous process</a>, i.e., for <a href="Almost_all" title="Almost all">almost all</a> realisations Bt(ω) of the noise, Xt(ω) is a <a href="Continuous_function" title="Continuous function">continuous function</a> of the time parameter, t. More accurately, there is a "continuous version" of X, a continuous process Y so that

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} [X_{t}=Y_{t}]=1{\mbox{ for all }}t.}</annotation>

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</math><img src="./dfadd74118eeaec23dfc31d658c76eb54b7e935a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.877ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} [X_{t}=Y_{t}]=1{\mbox{ for all }}t.}" loading="lazy"></dd></dl>

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

<div class="mw-heading mw-heading3"><h3 id="Feller_continuity">Feller continuity</h3></div>

In addition to being (sample) continuous, an Itô diffusion X satisfies the stronger requirement to be a <a href="Feller-continuous_process" title="Feller-continuous process">Feller-continuous process</a>.

For a point x ∈ Rn, let Px denote the law of X given initial datum X0 = x, and let Ex denote <a href="Expected_value" title="Expected value">expectation</a> with respect to Px.

Let f : Rn → R be a <a href="Borel_sigma_algebra" class="mw-redirect" title="Borel sigma algebra">Borel</a>-<a href="Measurable_function" title="Measurable function">measurable function</a> that is <a href="Bounded_function" title="Bounded function">bounded below</a> and define, for fixed t ≥ 0, u : Rn → R by

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<annotation encoding="application/x-tex">{\displaystyle u(x)=\mathbf {E} ^{x}[f(X_{t})].}</annotation>

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</math><img src="./67074cef3f266b25d7bc717b0fcfd8d4d340a24d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.275ex; height:2.843ex;" alt="{\displaystyle u(x)=\mathbf {E} ^{x}[f(X_{t})].}" loading="lazy"></dd></dl>

<ul><li><a href="Semi-continuity" title="Semi-continuity">Lower semi-continuity</a>: if f is lower semi-continuous, then u is lower semi-continuous.</li>

<li>Feller continuity: if f is bounded and continuous, then u is continuous.</li></ul>

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.)

<div class="mw-heading mw-heading2"><h2 id="The_Markov_property">The Markov property</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Markov_property" title="Markov property">Markov property</a></div>

<div class="mw-heading mw-heading3"><h3 id="The_Markov_property_2">The Markov property</h3></div>

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 <a href="Natural_filtration" title="Natural filtration">natural</a> <a href="Filtration_(abstract_algebra)" class="mw-redirect" title="Filtration (abstract algebra)">filtration</a> of (Ω, Σ) generated by the Brownian motion B: for t ≥ 0,

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<mtext> Borel</mtext>

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<annotation encoding="application/x-tex">{\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\}.}</annotation>

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</math><img src="./491e787787645b67cf1362b2d5f23c586330b212.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:57.21ex; height:3.343ex;" alt="{\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\}.}" loading="lazy"></dd></dl>

It is easy to show that X is <a href="Adapted_process" title="Adapted process">adapted</a> to Σ∗ (i.e. each Xt is Σt-measurable), so the natural filtration F∗ = F∗X of (Ω, Σ) 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 <a href="Conditional_expectation" title="Conditional expectation">conditional expectation</a> conditioned on the <a href="Sigma-algebra" class="mw-redirect" title="Sigma-algebra">σ-algebra</a> Σt and the expectation of the process "restarted" from Xt satisfy the Markov property:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ^{x}{\big [}f(X_{t+h}){\big |}\Sigma _{t}{\big ]}(\omega )=\mathbf {E} ^{X_{t}(\omega )}[f(X_{h})].}</annotation>

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</math><img src="./8903fa0e80708853aeb75a0263297befd5193e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.866ex; height:3.509ex;" alt="{\displaystyle \mathbf {E} ^{x}{\big [}f(X_{t+h}){\big |}\Sigma _{t}{\big ]}(\omega )=\mathbf {E} ^{X_{t}(\omega )}[f(X_{h})].}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./51249c7b389df0130d40c46927cb4e83229e27bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:45.218ex; height:10.176ex;" alt="{\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}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="The_strong_Markov_property">The strong Markov property</h3></div>

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 <a href="Stopping_time" title="Stopping time">stopping time</a>. 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 < +∞ <a href="Almost_surely" title="Almost surely">almost surely</a>. Then, for all h ≥ 0,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ^{x}{\big [}f(X_{\tau +h}){\big |}\Sigma _{\tau }{\big ]}=\mathbf {E} ^{X_{\tau }}{\big [}f(X_{h}){\big ]}.}</annotation>

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</math><img src="./76020ea9a111d51812ca2dc4539437372f2aeda8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.676ex; height:3.343ex;" alt="{\displaystyle \mathbf {E} ^{x}{\big [}f(X_{\tau +h}){\big |}\Sigma _{\tau }{\big ]}=\mathbf {E} ^{X_{\tau }}{\big [}f(X_{h}){\big ]}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="The_generator">The generator</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator (stochastic processes)</a></div>

<div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3></div>

Associated to each Itô diffusion, there is a second-order <a href="Partial_differential_operator" class="mw-redirect" title="Partial differential operator">partial differential operator</a> 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

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<annotation encoding="application/x-tex">{\displaystyle Af(x)=\lim _{t\downarrow 0}{\frac {\mathbf {E} ^{x}[f(X_{t})]-f(x)}{t}}.}</annotation>

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</math><img src="./fe0bbacf1a9e7e584f321a9a32fd87f6218776b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.678ex; height:6.176ex;" alt="{\displaystyle Af(x)=\lim _{t\downarrow 0}{\frac {\mathbf {E} ^{x}[f(X_{t})]-f(x)}{t}}.}" loading="lazy"></dd></dl>

The set of all functions f for which this limit exists at a point x is denoted DA(x), while DA denotes the set of all f for which the limit exists for all x ∈ Rn. One can show that any <a href="Compact_support" class="mw-redirect" title="Compact support">compactly-supported</a> C2 (twice differentiable with continuous second derivative) function f lies in DA and that

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<annotation encoding="application/x-tex">{\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),}</annotation>

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</math><img src="./2ee84914d15f5915b2bff0e616b6f9a38bb98575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:60.514ex; height:7.176ex;" alt="{\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),}" loading="lazy"></dd></dl>

or, in terms of the <a href="Gradient" title="Gradient">gradient</a> and <a href="Dot_product" title="Dot product">scalar</a> and <a href="Frobenius_inner_product" title="Frobenius inner product">Frobenius</a> <a href="Inner_product" class="mw-redirect" title="Inner product">inner products</a>,

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<annotation encoding="application/x-tex">{\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).}</annotation>

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</math><img src="./25cadd39e25ca314dac2e6bf544044f77ca5f47e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:53.283ex; height:3.509ex;" alt="{\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).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="An_example">An example</h3></div>

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

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<annotation encoding="application/x-tex">{\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)}</annotation>

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</math><img src="./d208f6ae9723f5b4c51c6052c04aafc31b261c60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:45.486ex; height:7.176ex;" alt="{\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)}" loading="lazy">,</dd></dl>

i.e., A = Δ/2, where Δ denotes the <a href="Laplace_operator" title="Laplace operator">Laplace operator</a>.

<div class="mw-heading mw-heading3"><h3 id="The_Kolmogorov_and_Fokker–Planck_equations">The Kolmogorov and Fokker–Planck equations</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Kolmogorov_backward_equation" class="mw-redirect" title="Kolmogorov backward equation">Kolmogorov backward equation</a> and <a href="Fokker%E2%80%93Planck_equation" title="Fokker–Planck equation">Fokker–Planck equation</a></div>

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 <a href="Partial_differential_equation" title="Partial differential equation">partial differential equation</a> 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

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<annotation encoding="application/x-tex">{\displaystyle u(t,x)=\mathbf {E} ^{x}[f(X_{t})],}</annotation>

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</math><img src="./55d3e9ef1ecad2266c26e7154237233328c88e99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.149ex; height:2.843ex;" alt="{\displaystyle u(t,x)=\mathbf {E} ^{x}[f(X_{t})],}" loading="lazy"></dd></dl>

then u(t, x) is differentiable with respect to t, u(t, ·) ∈ DA for all t, and u satisfies the following <a href="Partial_differential_equation" title="Partial differential equation">partial differential equation</a>, known as Kolmogorov's backward equation:

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}}">

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./96f34a768e02e9da4a3a2ffa6e1754f008ce051f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.37ex; margin-bottom: -0.301ex; width:39.64ex; height:8.509ex;" alt="{\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}}}" loading="lazy"></dd></dl>

The Fokker–Planck equation (also known as Kolmogorov's forward equation) is in some sense the "<a href="Adjoint" title="Adjoint">adjoint</a>" to the backward equation, and tells us how the <a href="Probability_density_function" title="Probability density function">probability density functions</a> of Xt evolve with time t. Let ρ(t, ·) be the density of Xt with respect to <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> on Rn, i.e., for any Borel-measurable set S ⊆ Rn,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} \left[X_{t}\in S\right]=\int _{S}\rho (t,x)\,\mathrm {d} x.}</annotation>

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</math><img src="./50cbb30dfea0eb40f642d7706d4f99f5839e1d48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.539ex; height:5.676ex;" alt="{\displaystyle \mathbf {P} \left[X_{t}\in S\right]=\int _{S}\rho (t,x)\,\mathrm {d} x.}" loading="lazy"></dd></dl>

Let A∗ denote the <a href="Hermitian_adjoint" title="Hermitian adjoint">Hermitian adjoint</a> of A (with respect to the <a href="Lp_space" title="Lp space">L2</a> <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a>). Then, given that the initial position X0 has a prescribed density ρ0, ρ(t, x) is differentiable with respect to t, ρ(t, ·) ∈ DA* for all t, and ρ satisfies the following partial differential equation, known as the Fokker–Planck equation:

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}}">

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./bdf73115b02439155fc6600d9a7b1285ce7ed451.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:40.439ex; height:8.509ex;" alt="{\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}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="The_Feynman–Kac_formula">The Feynman–Kac formula</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></div>

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 <a href="Continuous_function" title="Continuous function">continuous function</a> that is bounded below. Define a function v : [0, +∞) × Rn → R by

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<annotation encoding="application/x-tex">{\displaystyle v(t,x)=\mathbf {E} ^{x}\left[\exp \left(-\int _{0}^{t}q(X_{s})\,\mathrm {d} s\right)f(X_{t})\right].}</annotation>

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</math><img src="./f987cda421266eb13c9699a2fd7098704cb7698a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.305ex; height:6.343ex;" alt="{\displaystyle v(t,x)=\mathbf {E} ^{x}\left[\exp \left(-\int _{0}^{t}q(X_{s})\,\mathrm {d} s\right)f(X_{t})\right].}" loading="lazy"></dd></dl>

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

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}}">

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./6361b2c9a17dce619f6b92652e43e4574b621a57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.37ex; margin-bottom: -0.301ex; width:52.425ex; height:8.509ex;" alt="{\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}}}" loading="lazy"></dd></dl>

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.

<div class="mw-heading mw-heading2"><h2 id="The_characteristic_operator">The characteristic operator</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition_2">Definition</h3></div>

The <a href="Characteristic_operator" class="mw-redirect" title="Characteristic operator">characteristic operator</a> 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 <a href="Dirichlet_problem" title="Dirichlet problem">Dirichlet problem</a>.

The characteristic operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

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</math><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"> of an Itô diffusion X is defined by

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<annotation encoding="application/x-tex">{\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}]}},}</annotation>

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</math><img src="./082b14ed2c141dc8165a21107f2913bffea6b658.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.183ex; height:6.843ex;" alt="{\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}]}},}" loading="lazy"></dd></dl>

where the sets U form a sequence of <a href="Open_set" title="Open set">open sets</a> Uk that decrease to the point x in the sense that

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<annotation encoding="application/x-tex">{\displaystyle U_{k+1}\subseteq U_{k}{\mbox{ and }}\bigcap _{k=1}^{\infty }U_{k}=\{x\},}</annotation>

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</math><img src="./c7c7a96a5945f4897eaa39ec1631b40140d50350.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.268ex; height:6.843ex;" alt="{\displaystyle U_{k+1}\subseteq U_{k}{\mbox{ and }}\bigcap _{k=1}^{\infty }U_{k}=\{x\},}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle \tau _{U}=\inf\{t\geq 0\ :\ X_{t}\not \in U\}}</annotation>

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</math><img src="./97464921f7e9003365eae816ce68adde56c4c9ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.156ex; height:2.843ex;" alt="{\displaystyle \tau _{U}=\inf\{t\geq 0\ :\ X_{t}\not \in U\}}" loading="lazy"></dd></dl>

is the first exit time from U for X. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mathcal {A}}}">

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</math><img src="./17c62c6c23c0ac19891d206b64a4cc8853ecccf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.502ex; height:2.509ex;" alt="{\displaystyle D_{\mathcal {A}}}" loading="lazy"> denotes the set of all f for which this limit exists for all x ∈ Rn and all sequences {Uk}. If Ex[τU] = +∞ for all open sets U containing x, define

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}f(x)=0.}</annotation>

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</math><img src="./1383740490171465bde62b37fad3f694440b74ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.229ex; height:2.843ex;" alt="{\displaystyle {\mathcal {A}}f(x)=0.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Relationship_with_the_generator">Relationship with the generator</h3></div>

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

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<annotation encoding="application/x-tex">{\displaystyle D_{A}\subseteq D_{\mathcal {A}}}</annotation>

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</math><img src="./5285d0f348288a749675a75b447f13da32ebb01f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.99ex; height:2.509ex;" alt="{\displaystyle D_{A}\subseteq D_{\mathcal {A}}}" loading="lazy"></dd></dl>

and that

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<annotation encoding="application/x-tex">{\displaystyle Af={\mathcal {A}}f{\mbox{ for all }}f\in D_{A}.}</annotation>

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</math><img src="./459266a8ab9afd824458e9052e6a8306b6889508.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.441ex; height:2.676ex;" alt="{\displaystyle Af={\mathcal {A}}f{\mbox{ for all }}f\in D_{A}.}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\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).}</annotation>

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</math><img src="./1ddc003165b8961e85ab3d93668464c833843b2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:60.674ex; height:7.176ex;" alt="{\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).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Application:_Brownian_motion_on_a_Riemannian_manifold">Application: Brownian motion on a Riemannian manifold</h3></div>

Above, the generator (and hence characteristic operator) of Brownian motion on Rn was calculated to be ⁠1/2⁠Δ, where Δ denotes the Laplace operator. The characteristic operator is useful in defining Brownian motion on an m-dimensional <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> (M, g): a Brownian motion on M is defined to be a diffusion on M whose characteristic operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

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</math><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"> in local coordinates xi, 1 ≤ i ≤ m, is given by ⁠1/2⁠ΔLB, where ΔLB is the <a href="Laplace_operator#Laplace–Beltrami_operator" title="Laplace operator">Laplace-Beltrami operator</a> given in local coordinates by

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<annotation encoding="application/x-tex">{\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),}</annotation>

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</math><img src="./82d2a5b50947bb36cb6627c12e214fe596560dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:49.884ex; height:7.676ex;" alt="{\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),}" loading="lazy"></dd></dl>

where [gij] = [gij]−1 in the sense of <a href="Invertible_matrix" title="Invertible matrix">the inverse of a square matrix</a>.

<div class="mw-heading mw-heading2"><h2 id="The_resolvent_operator">The resolvent operator</h2></div>

In general, the generator A of an Itô diffusion X is not a <a href="Bounded_operator" title="Bounded operator">bounded operator</a>. 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 <a href="Feller_process#Resolvent" title="Feller process">resolvent</a> operator.

For α > 0, the resolvent operator Rα, acting on bounded, continuous functions g : Rn → R, is defined by

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<annotation encoding="application/x-tex">{\displaystyle R_{\alpha }g(x)=\mathbf {E} ^{x}\left[\int _{0}^{\infty }e^{-\alpha t}g(X_{t})\,\mathrm {d} t\right].}</annotation>

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</math><img src="./56d5db5ad58b18eedbf3ea67553af0caec488d08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.368ex; height:6.176ex;" alt="{\displaystyle R_{\alpha }g(x)=\mathbf {E} ^{x}\left[\int _{0}^{\infty }e^{-\alpha t}g(X_{t})\,\mathrm {d} t\right].}" loading="lazy"></dd></dl>

It can be shown, using the Feller continuity of the diffusion X, that Rαg is itself a bounded, continuous function. Also, Rα and αI − A are mutually inverse operators:

<ul><li>if f : Rn → R is C2 with compact support, then, for all α > 0,</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle R_{\alpha }(\alpha \mathbf {I} -A)f=f;}</annotation>

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</math><img src="./c7846cbabf960ca33615c62b308cbb2f9f4e4d9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.245ex; height:2.843ex;" alt="{\displaystyle R_{\alpha }(\alpha \mathbf {I} -A)f=f;}" loading="lazy"></dd></dl></dd></dl>

<ul><li>if g : Rn → R is bounded and continuous, then Rαg lies in DA and, for all α > 0,</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle (\alpha \mathbf {I} -A)R_{\alpha }g=g.}</annotation>

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</math><img src="./d085f53d7d7eb09bd87d37481edbc162db9d957c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.92ex; height:2.843ex;" alt="{\displaystyle (\alpha \mathbf {I} -A)R_{\alpha }g=g.}" loading="lazy"></dd></dl></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Invariant_measures">Invariant measures</h2></div>

Sometimes it is necessary to find an <a href="Invariant_measure" title="Invariant measure">invariant measure</a> for an Itô diffusion X, i.e. a measure on Rn that does not change under the "flow" of X: i.e., if X0 is distributed according to such an invariant measure μ∞, then Xt 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 X0 is indeed distributed according to an invariant measure μ∞ with density ρ∞, then the density ρ(t, ·) of Xt does not change with t, so ρ(t, ·) = ρ∞, and so ρ∞ must solve the (time-independent) partial differential equation

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<annotation encoding="application/x-tex">{\displaystyle A^{*}\rho _{\infty }(x)=0,\quad x\in \mathbf {R} ^{n}.}</annotation>

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</math><img src="./1eb710025cfb0f3252daead62f4164f6ce9b1030.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.67ex; height:2.843ex;" alt="{\displaystyle A^{*}\rho _{\infty }(x)=0,\quad x\in \mathbf {R} ^{n}.}" loading="lazy"></dd></dl>

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 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.

<div class="mw-heading mw-heading3"><h3 id="Invariant_measures_for_gradient_flows">Invariant measures for gradient flows</h3></div>

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

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} X_{t}=-\nabla \Psi (X_{t})\,\mathrm {d} t+{\sqrt {2\beta ^{-1}}}\,\mathrm {d} B_{t},}</annotation>

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</math><img src="./cfff73698b91d37929670ca5044b5acf64adadf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:34.685ex; height:4.843ex;" alt="{\displaystyle \mathrm {d} X_{t}=-\nabla \Psi (X_{t})\,\mathrm {d} t+{\sqrt {2\beta ^{-1}}}\,\mathrm {d} B_{t},}" loading="lazy"></dd></dl>

where β > 0 plays the role of an <a href="Inverse_temperature" class="mw-redirect" title="Inverse temperature">inverse temperature</a> 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 <a href="Gibbs_distribution" class="mw-redirect" title="Gibbs distribution">Gibbs distribution</a>:

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<annotation encoding="application/x-tex">{\displaystyle \rho _{\infty }(x)=Z^{-1}\exp(-\beta \Psi (x)),}</annotation>

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</math><img src="./d98366fd397f26cadd309879aa5cd2cf798e59db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.839ex; height:3.176ex;" alt="{\displaystyle \rho _{\infty }(x)=Z^{-1}\exp(-\beta \Psi (x)),}" loading="lazy"></dd></dl>

where the <a href="Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a> Z is given by

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<annotation encoding="application/x-tex">{\displaystyle Z=\int _{\mathbf {R} ^{n}}\exp(-\beta \Psi (x))\,\mathrm {d} x.}</annotation>

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</math><img src="./3b244da4989e664d399a8ec15ea324cbc82938cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.177ex; height:5.676ex;" alt="{\displaystyle Z=\int _{\mathbf {R} ^{n}}\exp(-\beta \Psi (x))\,\mathrm {d} x.}" loading="lazy"></dd></dl>

Moreover, the density ρ∞ satisfies a <a href="Variational_principle" title="Variational principle">variational principle</a>: it minimizes over all probability densities ρ on Rn the <a href="Thermodynamic_free_energy" title="Thermodynamic free energy">free energy</a> functional F given by

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<annotation encoding="application/x-tex">{\displaystyle F[\rho ]=E[\rho ]+{\frac {1}{\beta }}S[\rho ],}</annotation>

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</math><img src="./f27ebff00cf0748b5abf723416b9c7f883d2c5e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.256ex; height:5.676ex;" alt="{\displaystyle F[\rho ]=E[\rho ]+{\frac {1}{\beta }}S[\rho ],}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle E[\rho ]=\int _{\mathbf {R} ^{n}}\Psi (x)\rho (x)\,\mathrm {d} x}</annotation>

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</math><img src="./b462ab6458616932b57a68bc30ea66c30db3915c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.96ex; height:5.676ex;" alt="{\displaystyle E[\rho ]=\int _{\mathbf {R} ^{n}}\Psi (x)\rho (x)\,\mathrm {d} x}" loading="lazy"></dd></dl>

plays the role of an energy functional, and

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<annotation encoding="application/x-tex">{\displaystyle S[\rho ]=\int _{\mathbf {R} ^{n}}\rho (x)\log \rho (x)\,\mathrm {d} x}</annotation>

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</math><img src="./1a99acc454700129da879ee0b708035dd93ae900.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.824ex; height:5.676ex;" alt="{\displaystyle S[\rho ]=\int _{\mathbf {R} ^{n}}\rho (x)\log \rho (x)\,\mathrm {d} x}" loading="lazy"></dd></dl>

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 <a href="Lyapunov_function" title="Lyapunov function">Lyapunov function</a> for the Fokker–Planck equation: F[ρ(t, ·)] must decrease as t increases. Thus, F is an <a href="H-theorem" title="H-theorem">H-function</a> for the X-dynamics.

<div class="mw-heading mw-heading3"><h3 id="Example">Example</h3></div>

Consider the <a href="Ornstein-Uhlenbeck_process" class="mw-redirect" title="Ornstein-Uhlenbeck process">Ornstein-Uhlenbeck process</a> X on Rn satisfying the stochastic differential equation

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} X_{t}=-\kappa (X_{t}-m)\,\mathrm {d} t+{\sqrt {2\beta ^{-1}}}\,\mathrm {d} B_{t},}</annotation>

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</math><img src="./db02dfde8517935e36d11cece4543e5d6a9dcce0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:37.161ex; height:4.843ex;" alt="{\displaystyle \mathrm {d} X_{t}=-\kappa (X_{t}-m)\,\mathrm {d} t+{\sqrt {2\beta ^{-1}}}\,\mathrm {d} B_{t},}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle \Psi (x)={\tfrac {1}{2}}\kappa |x-m|^{2},}</annotation>

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</math><img src="./a3b5b660d8a33ae0de7d5d6b1acbffb70fa181f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.248ex; height:3.676ex;" alt="{\displaystyle \Psi (x)={\tfrac {1}{2}}\kappa |x-m|^{2},}" loading="lazy"></dd></dl>

and so the invariant measure for X is a <a href="Gaussian_measure" title="Gaussian measure">Gaussian measure</a> with density ρ∞ given by

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<annotation encoding="application/x-tex">{\displaystyle \rho _{\infty }(x)=\left({\frac {\beta \kappa }{2\pi }}\right)^{\frac {n}{2}}\exp \left(-{\frac {\beta \kappa |x-m|^{2}}{2}}\right)}</annotation>

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</math><img src="./e3568e80e4824f6063583e226aede01424d9ee2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.607ex; height:7.676ex;" alt="{\displaystyle \rho _{\infty }(x)=\left({\frac {\beta \kappa }{2\pi }}\right)^{\frac {n}{2}}\exp \left(-{\frac {\beta \kappa |x-m|^{2}}{2}}\right)}" loading="lazy">.</dd></dl>

Heuristically, for large t, Xt is approximately <a href="Normal_distribution" title="Normal distribution">normally distributed</a> 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 Xt 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, Xt is unlikely to move far away from m.

<div class="mw-heading mw-heading2"><h2 id="The_martingale_property">The martingale property</h2></div>

In general, an Itô diffusion X is not a <a href="Martingale_(probability_theory)" title="Martingale (probability theory)">martingale</a>. However, for any f ∈ C2(Rn; R) with compact support, the process M : [0, +∞) × Ω → R defined by

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<annotation encoding="application/x-tex">{\displaystyle M_{t}=f(X_{t})-\int _{0}^{t}Af(X_{s})\,\mathrm {d} s,}</annotation>

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</math><img src="./2888b91e564a3d06861c7a7fdafade68a5ae5d72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.71ex; height:6.176ex;" alt="{\displaystyle M_{t}=f(X_{t})-\int _{0}^{t}Af(X_{s})\,\mathrm {d} s,}" loading="lazy"></dd></dl>

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 <a href="It%C3%B4's_lemma" title="Itô's lemma">Itô's lemma</a> (the stochastic <a href="Chain_rule" title="Chain rule">chain rule</a>) that

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<annotation encoding="application/x-tex">{\displaystyle f(X_{t})=f(x)+\int _{0}^{t}Af(X_{s})\,\mathrm {d} s+\int _{0}^{t}\nabla f(X_{s})^{\top }\sigma (X_{s})\,\mathrm {d} B_{s}.}</annotation>

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</math><img src="./e4b53370188ce5f70bef2937a1432befafeb06a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:57.541ex; height:6.176ex;" alt="{\displaystyle f(X_{t})=f(x)+\int _{0}^{t}Af(X_{s})\,\mathrm {d} s+\int _{0}^{t}\nabla f(X_{s})^{\top }\sigma (X_{s})\,\mathrm {d} B_{s}.}" loading="lazy"></dd></dl>

Since Itô integrals are martingales with respect to the natural filtration Σ∗ of (Ω, Σ) by B, for t > s,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ^{x}{\big [}M_{t}{\big |}\Sigma _{s}{\big ]}=M_{s}.}</annotation>

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</math><img src="./f240c98d34be95ede8af2b8346f9bbc75bc193a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.28ex; height:3.176ex;" alt="{\displaystyle \mathbf {E} ^{x}{\big [}M_{t}{\big |}\Sigma _{s}{\big ]}=M_{s}.}" loading="lazy"></dd></dl>

Hence, as required,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ^{x}[M_{t}|F_{s}]=\mathbf {E} ^{x}\left[\mathbf {E} ^{x}{\big [}M_{t}{\big |}\Sigma _{s}{\big ]}{\big |}F_{s}\right]=\mathbf {E} ^{x}{\big [}M_{s}{\big |}F_{s}{\big ]}=M_{s},}</annotation>

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</math><img src="./64fb86554a6af76f82df2f389932736094e0898f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:54.597ex; height:3.176ex;" alt="{\displaystyle \mathbf {E} ^{x}[M_{t}|F_{s}]=\mathbf {E} ^{x}\left[\mathbf {E} ^{x}{\big [}M_{t}{\big |}\Sigma _{s}{\big ]}{\big |}F_{s}\right]=\mathbf {E} ^{x}{\big [}M_{s}{\big |}F_{s}{\big ]}=M_{s},}" loading="lazy"></dd></dl>

since Ms is Fs-measurable.

<div class="mw-heading mw-heading2"><h2 id="Dynkin's_formula">Dynkin's formula</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Dynkin's_formula" title="Dynkin's formula">Dynkin's formula</a></div>

Dynkin's formula, named after <a href="Eugene_Dynkin" title="Eugene Dynkin">Eugene Dynkin</a>, gives the <a href="Expected_value" title="Expected value">expected value</a> of any suitably smooth statistic of an Itô diffusion X (with generator A) at a stopping time. Precisely, if τ is a stopping time with Ex[τ] < +∞, and f : Rn → R is C2 with compact support, then

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ^{x}[f(X_{\tau })]=f(x)+\mathbf {E} ^{x}\left[\int _{0}^{\tau }Af(X_{s})\,\mathrm {d} s\right].}</annotation>

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</math><img src="./40310c2ff87077bbda2c6656e3ba340b6b70feb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.942ex; height:6.176ex;" alt="{\displaystyle \mathbf {E} ^{x}[f(X_{\tau })]=f(x)+\mathbf {E} ^{x}\left[\int _{0}^{\tau }Af(X_{s})\,\mathrm {d} s\right].}" loading="lazy"></dd></dl>

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 <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> (−R, +R) at a random time τR with expected value

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ^{0}[\tau _{R}]=R^{2}.}</annotation>

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</math><img src="./e47ef7b9385e4da453f13672e41e84515392c0b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.164ex; height:3.176ex;" alt="{\displaystyle \mathbf {E} ^{0}[\tau _{R}]=R^{2}.}" loading="lazy"></dd></dl>

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 <a href="Hitting_time" title="Hitting time">hitting time</a>, one can study the harmonic measure of the process.

<div class="mw-heading mw-heading2"><h2 id="Associated_measures">Associated measures</h2></div>

<div class="mw-heading mw-heading3"><h3 id="The_harmonic_measure">The harmonic measure</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Harmonic_measure" title="Harmonic measure">Harmonic measure</a></div>

In many situations, it is sufficient to know when an Itô diffusion X will first leave a <a href="Measurable_set" class="mw-redirect" title="Measurable set">measurable set</a> H ⊆ Rn. That is, one wishes to study the <a href="Hitting_time" title="Hitting time">first exit time</a>

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<annotation encoding="application/x-tex">{\displaystyle \tau _{H}(\omega )=\inf\{t\geq 0|X_{t}\not \in H\}.}</annotation>

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</math><img src="./28c9e75d60780eccadff40be6fdaa2deecb508ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.086ex; height:2.843ex;" alt="{\displaystyle \tau _{H}(\omega )=\inf\{t\geq 0|X_{t}\not \in H\}.}" loading="lazy"></dd></dl>

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 <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> (−1, 1) at −1 with probability ⁠1/2⁠ and at 1 with probability ⁠1/2⁠, so Bτ(−1, 1) is <a href="Uniform_distribution_(discrete)" class="mw-redirect" title="Uniform distribution (discrete)">uniformly distributed</a> on the set {−1, 1}.

In general, if G is <a href="Compactly_embedded" class="mw-redirect" title="Compactly embedded">compactly embedded</a> within Rn, then the harmonic measure (or hitting distribution) of X on the <a href="Boundary_(topology)" title="Boundary (topology)">boundary</a> ∂G of G is the measure μGx defined by

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<annotation encoding="application/x-tex">{\displaystyle \mu _{G}^{x}(F)=\mathbf {P} ^{x}\left[X_{\tau _{G}}\in F\right]}</annotation>

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</math><img src="./3d9281fd0d7a7d620dde6c7a7edabe00be1bb67f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.569ex; height:3.176ex;" alt="{\displaystyle \mu _{G}^{x}(F)=\mathbf {P} ^{x}\left[X_{\tau _{G}}\in F\right]}" loading="lazy"></dd></dl>

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 <a href="Open_ball" class="mw-redirect" title="Open ball">open ball</a> centred on x, then the harmonic measure of B on ∂D is <a href="Invariant_measure" title="Invariant measure">invariant</a> under all <a href="Rotation" title="Rotation">rotations</a> 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

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<annotation encoding="application/x-tex">{\displaystyle \varphi (x)=\mathbf {E} ^{x}\left[f(X_{\tau _{H}})\right],}</annotation>

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</math><img src="./868e6fbcc7436e4d657b6f54bbbf258a7b3ec88f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.714ex; height:3.009ex;" alt="{\displaystyle \varphi (x)=\mathbf {E} ^{x}\left[f(X_{\tau _{H}})\right],}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle \varphi (x)=\int _{\partial G}\varphi (y)\,\mathrm {d} \mu _{G}^{x}(y).}</annotation>

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</math><img src="./df98efffdf6cf2820d5d61199d723e7f3f78e656.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.595ex; height:5.676ex;" alt="{\displaystyle \varphi (x)=\int _{\partial G}\varphi (y)\,\mathrm {d} \mu _{G}^{x}(y).}" loading="lazy"></dd></dl>

The mean value property is very useful in the <a href="Stochastic_processes_and_boundary_value_problems" title="Stochastic processes and boundary value problems">solution of partial differential equations using stochastic processes</a>.

<div class="mw-heading mw-heading3"><h3 id="The_Green_measure_and_Green_formula">The Green measure and Green formula</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Green_measure" title="Green measure">Green measure</a></div>

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

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<annotation encoding="application/x-tex">{\displaystyle G(x,H)=\mathbf {E} ^{x}\left[\int _{0}^{\tau _{D}}\chi _{H}(X_{s})\,\mathrm {d} s\right],}</annotation>

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</math><img src="./c4c31b6290036aab7780f2db087113cf571f14d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.692ex; height:6.176ex;" alt="{\displaystyle G(x,H)=\mathbf {E} ^{x}\left[\int _{0}^{\tau _{D}}\chi _{H}(X_{s})\,\mathrm {d} s\right],}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\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].}</annotation>

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</math><img src="./9d8a58f4f52c4cdc917d0f5f662c31728b358ec5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.111ex; height:6.176ex;" alt="{\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].}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle G(x,H)=\int _{H}G(x,y)\,\mathrm {d} y,}</annotation>

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</math><img src="./b73938c1425268459b2fb975e38c74509180209e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.17ex; height:5.676ex;" alt="{\displaystyle G(x,H)=\int _{H}G(x,y)\,\mathrm {d} y,}" loading="lazy"></dd></dl>

where G(x, y) is <a href="Green's_function" title="Green's function">Green's function</a> for the operator ⁠1/2⁠Δ on the domain D.

Suppose that Ex[τD] < +∞ for all x ∈ D. Then the Green formula holds for all f ∈ C2(Rn; R) with compact support:

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<annotation encoding="application/x-tex">{\displaystyle f(x)=\mathbf {E} ^{x}\left[f\left(X_{\tau _{D}}\right)\right]-\int _{D}Af(y)\,G(x,\mathrm {d} y).}</annotation>

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</math><img src="./e2e0ff4f260285924d550fea6f23411ef3c6504b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.326ex; height:5.676ex;" alt="{\displaystyle f(x)=\mathbf {E} ^{x}\left[f\left(X_{\tau _{D}}\right)\right]-\int _{D}Af(y)\,G(x,\mathrm {d} y).}" loading="lazy"></dd></dl>

In particular, if the support of f is <a href="Compactly_embedded" class="mw-redirect" title="Compactly embedded">compactly embedded</a> in D,

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<annotation encoding="application/x-tex">{\displaystyle f(x)=-\int _{D}Af(y)\,G(x,\mathrm {d} y).}</annotation>

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</math><img src="./6f93abec7c69f7b32f7e7a77d543584af7068d74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.452ex; height:5.676ex;" alt="{\displaystyle f(x)=-\int _{D}Af(y)\,G(x,\mathrm {d} y).}" loading="lazy"></dd></dl>

<ul><li><a href="Diffusion_process" title="Diffusion process">Diffusion process</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFDynkintrans._J._FabiusV._GreenbergA._Maitra1965" class="citation book cs1"><a href="Eugene_Dynkin" title="Eugene Dynkin">Dynkin, Eugene B.</a>; 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.</cite> <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0193671">0193671</a></li>

<li><cite id="CITEREFJordanKinderlehrer,_DavidOtto,_Felix1998" class="citation journal cs1">Jordan, Richard; Kinderlehrer, David; <a href="Felix_Otto_(mathematician)" title="Felix Otto (mathematician)">Otto, Felix</a> (1998). "The variational formulation of the Fokker–Planck equation". SIAM J. Math. Anal. 29 (1): 1–17 (electronic). <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.6.8815">10.1.1.6.8815</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0036141096303359">10.1137/S0036141096303359</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13890235">13890235</a>.</cite> <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1617171">1617171</a></li>

<li><cite id="CITEREFØksendal2003" class="citation book cs1"><a href="Bernt_%C3%98ksendal" title="Bernt Øksendal">Øksendal, Bernt K.</a> (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-04758-1</bdi>.</cite> <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2001996">2001996</a> (See Sections 7, 8 and 9)</li></ul>

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</style><div id="Stochastic_processes496" style="font-size:114%;margin:0 4em"><a href="Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>

<li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li>

<li><a href="Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li>

<li><a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li>

<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>

<li><a href="Moran_process" title="Moran process">Moran process</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a>

<ul><li><a href="Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li>

<li><a href="Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li>

<li><a href="Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li>

<li><a href="Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Additive_process" title="Additive process">Additive process</a></li>

<li><a href="Airy_process" title="Airy process">Airy process</a></li>

<li><a href="Bessel_process" title="Bessel process">Bessel process</a></li>

<li><a href="Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a>

<ul><li><a href="Birth_process" title="Birth process">pure birth</a></li></ul></li>

<li><a href="Wiener_process" title="Wiener process">Brownian motion</a>

<ul><li><a href="Brownian_bridge" title="Brownian bridge">Bridge</a></li>

<li><a href="Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson</a></li>

<li><a href="Brownian_excursion" title="Brownian excursion">Excursion</a></li>

<li><a href="Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li>

<li><a href="Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li>

<li><a href="Brownian_meander" title="Brownian meander">Meander</a></li></ul></li>

<li><a href="Cauchy_process" title="Cauchy process">Cauchy process</a></li>

<li><a href="Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li>

<li><a href="Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li>

<li><a href="Cox_process" title="Cox process">Cox process</a></li>

<li><a href="Diffusion_process" title="Diffusion process">Diffusion process</a></li>

<li><a href="Empirical_process" title="Empirical process">Empirical process</a></li>

<li><a href="Feller_process" title="Feller process">Feller process</a></li>

<li><a href="Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li>

<li><a href="Gamma_process" title="Gamma process">Gamma process</a></li>

<li><a href="Geometric_process" title="Geometric process">Geometric process</a></li>

<li><a href="Hawkes_process" title="Hawkes process">Hawkes process</a></li>

<li><a href="Hunt_process" title="Hunt process">Hunt process</a></li>

<li><a href="Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li>

<li><a href="It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li>

<li><a href="Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li>

<li><a href="Jump_process" title="Jump process">Jump process</a></li>

<li><a href="L%C3%A9vy_process" title="Lévy process">Lévy process</a></li>

<li><a href="Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li>

<li><a href="Markov_additive_process" title="Markov additive process">Markov additive process</a></li>

<li><a href="McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li>

<li><a href="Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson process</a>

<ul><li><a href="Compound_Poisson_process" title="Compound Poisson process">Compound</a></li>

<li><a href="Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li>

<li><a href="Quasimartingale" title="Quasimartingale">Quasimartingale</a></li>

<li><a href="Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li>

<li><a href="Semimartingale" title="Semimartingale">Semimartingale</a></li>

<li><a href="Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li>

<li><a href="Stable_process" title="Stable process">Stable process</a></li>

<li><a href="Superprocess" title="Superprocess">Superprocess</a></li>

<li><a href="Telegraph_process" title="Telegraph process">Telegraph process</a></li>

<li><a href="Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li>

<li><a href="Wiener_process" title="Wiener process">Wiener process</a></li>

<li><a href="Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Gaussian_process" title="Gaussian process">Gaussian process</a></li>

<li><a href="Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li>

<li><a href="Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li>

<li><a href="Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a>

<ul><li><a href="Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li>

<li><a href="Local_martingale" title="Local martingale">Local</a></li>

<li><a href="Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li>

<li><a href="Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li>

<li><a href="Regenerative_process" title="Regenerative process">Regenerative process</a></li>

<li><a href="Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li>

<li><a href="Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li>

<li><a href="White_noise" title="White noise">White noise</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li>

<li><a href="Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li>

<li><a href="Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li>

<li><a href="Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li>

<li><a href="Ising_model" title="Ising model">Ising model</a>

<ul><li><a href="Potts_model" title="Potts model">Potts model</a></li>

<li><a href="Boolean_network" title="Boolean network">Boolean network</a></li></ul></li>

<li><a href="Markov_random_field" title="Markov random field">Markov random field</a></li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation</a></li>

<li><a href="Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li>

<li><a href="Point_process" title="Point process">Point process</a>

<li><a href="Determinantal_point_process" title="Determinantal point process">Determinantal</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li>

<li><a href="Random_field" title="Random field">Random field</a></li>

<li><a href="Random_graph" title="Random graph">Random graph</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li>

<li><a href="Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li>

<li><a href="Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li>

<li><a href="Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li>

<li><a href="Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li>

<li><a href="Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li>

<li><a href="Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li>

<li><a href="Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li>

<li><a href="Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li>

<li><a href="Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li>

<li><a href="Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li>

<li><a href="Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li>

<li><a href="Heston_model" title="Heston model">Heston</a></li>

<li><a href="Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li>

<li><a href="Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li>

<li><a href="LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li>

<li><a href="Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li>

<li><a href="SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li>

<li><a href="Vasicek_model" title="Vasicek model">Vašíček</a></li>

<li><a href="Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li>

<li><a href="Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li>

<li><a href="Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li>

<li><a href="Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Fluid_queue" title="Fluid queue">Fluid</a></li>

<li><a href="G-network" title="G-network">Generalized queueing network</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li>

<li><a href="Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li>

<li><a href="Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li>

<li><a href="Ergodicity" title="Ergodicity">Ergodic</a></li>

<li><a href="Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li>

<li><a href="Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li>

<li><a href="Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li>

<li><a href="Markov_property" title="Markov property">Markov</a></li>

<li><a href="Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li>

<li><a href="Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li>

<li><a href="Predictable_process" title="Predictable process">Predictable</a></li>

<li><a href="Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li>

<li><a href="Self-similar_process" title="Self-similar process">Self-similar</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationary</a></li>

<li><a href="Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Donsker's_theorem" title="Donsker's theorem">Donsker's theorem</a></li>

<li><a href="Doob's_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li>

<li><a href="Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li>

<li><a href="Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li>

<li><a href="Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li>

<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li>

<li><a href="Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li>

<li><a href="Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li>

<li><a href="Sanov's_theorem" title="Sanov's theorem">Sanov's theorem</a></li>

<li><a href="Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="Blumenthal's_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="Kolmogorov's_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="L%C3%A9vy's_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li>

<li><a href="Doob's_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li>

<li><a href="Doob's_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li>

<li><a href="Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li>

<li><a href="Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li>

<li><a href="Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li>

<li><a href="Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li>

<li><a href="Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li>

<li><a href="Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li>

<li><a href="Doob's_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li>

<li><a href="Dynkin's_formula" title="Dynkin's formula">Dynkin's formula</a></li>

<li><a href="Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li>

<li><a href="Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li>

<li><a href="Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li>

<li><a href="Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li>

<li><a href="It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li>

<li><a href="It%C3%B4's_lemma" title="Itô's lemma">Itô's lemma</a></li>

<li><a href="Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li>

<li><a href="Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li>

<li><a href="Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li>

<li><a href="L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li>

<li><a href="Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li>

<li><a href="Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li>

<li><a href="Prokhorov's_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li>

<li><a href="Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li>

<li><a href="Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li>

<li><a href="Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li>

<li><a href="Skorokhod's_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li>

<li><a href="Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li>

<li><a href="Snell_envelope" title="Snell envelope">Snell envelope</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a>

<ul><li><a href="Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li>

<li><a href="Stopping_time" title="Stopping time">Stopping time</a></li>

<li><a href="Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li>

<li><a href="Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li>

<li><a href="Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li>

<li><a href="Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a>

<ul><li><a href="Classical_Wiener_space" title="Classical Wiener space">Classical</a></li>

<li><a href="Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li>

<li><a href="Stochastic_control" title="Stochastic control">Control theory</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li>

<li><a href="Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li>

<li><a href="Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Queueing_theory" title="Queueing theory">Queueing theory</a></li>

<li><a href="Renewal_theory" title="Renewal theory">Renewal theory</a></li>

<li><a href="Ruin_theory" title="Ruin theory">Ruin theory</a></li>

<li><a href="Signal_processing" title="Signal processing">Signal processing</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a></li>

<li><a href="Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li>

<li><a href="Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li>

<li><a href="Machine_learning" title="Machine learning">Machine learning</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div>

<ul><li><a href="List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li>

<li>Category</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-06-20" href="https://en.wikipedia.org/wiki/?title=It%C3%B4_diffusion&oldid=1230019332">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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