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# Infinitesimal generator (stochastic processes)

In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is a partial differential operator that encodes a great deal of information about the process. The generator is used in evolution equations such as the Kolmogorov backward equation (which describes the evolution of statistics of the process); its L2 Hermitian adjoint is used in evolution equations such as the Fokker–Planck equation (which describes the evolution of the probability density functions of the process).[citation needed]

## Definition

### General case

For a Feller process ${\displaystyle (X_{t})_{t\geq 0}}$ we define the generator ${\displaystyle A}$ by

${\displaystyle Af(x):=\lim _{t\downarrow 0}{\frac {\mathbb {E} ^{x}(f(X_{t}))-f(x)}{t}}=\lim _{t\downarrow 0}{\frac {P_{t}f(x)-f(x)}{t}}}$

whenever this limit exists in ${\displaystyle (C_{\infty },\|\cdot \|_{\infty })}$.

### Stochastic differential equations driven by Brownian motion

Let ${\textstyle X:[0,\infty )\times \Omega \rightarrow \mathbb {R} ^{n}}$ defined on a probability space ${\textstyle (\Omega ,{\mathcal {F}},P)}$ be an Itô diffusion satisfying a stochastic differential equation of the form:

${\displaystyle \mathrm {d} X_{t}=b(X_{t})\,\mathrm {d} t+\sigma (X_{t})\,\mathrm {d} B_{t}}$

where ${\displaystyle B}$ is an m-dimensional Brownian motion and ${\displaystyle b:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}$ and ${\displaystyle \sigma :\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n\times m}}$ are the drift and diffusion fields respectively. For a point ${\displaystyle x\in \mathbb {R} ^{n}}$, let ${\displaystyle \mathbb {P} ^{x}}$ denote the law of ${\displaystyle X}$ given initial datum ${\displaystyle X_{0}=x}$, and let ${\displaystyle \mathbb {E} ^{x}}$ denote expectation with respect to ${\displaystyle \mathbb {P} ^{x}}$.

The infinitesimal generator of ${\displaystyle X}$ is the operator ${\displaystyle {\mathcal {A}}}$, which is defined to act on suitable functions ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ by:

${\displaystyle {\mathcal {A}}f(x)=\lim _{t\downarrow 0}{\frac {\mathbb {E} ^{x}[f(X_{t})]-f(x)}{t}}}$

The set of all functions ${\displaystyle f}$ for which this limit exists at a point ${\displaystyle x}$ is denoted ${\displaystyle D_{\mathcal {A}}(x)}$, while ${\displaystyle D_{\mathcal {A}}}$ denotes the set of all ${\displaystyle f}$ for which the limit exists for all ${\displaystyle x\in \mathbb {R} ^{n}}$. One can show that any compactly-supported ${\displaystyle C^{2}}$ (twice differentiable with continuous second derivative) function ${\displaystyle f}$ lies in ${\displaystyle D_{\mathcal {A}}}$ and that:

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

Or, in terms of the gradient and scalar and Frobenius inner products:

${\displaystyle {\mathcal {A}}f(x)=b(x)\cdot \nabla _{x}f(x)+{\frac {1}{2}}{\big (}\sigma (x)\sigma (x)^{\top }{\big )}:\nabla _{x}\nabla _{x}f(x)}$

## Generators of some common processes

• For finite-state continuous time Markov chains the generator may be expressed as a transition rate matrix
• Standard Brownian motion on ${\displaystyle \mathbb {R} ^{n}}$, which satisfies the stochastic differential equation ${\displaystyle dX_{t}=dB_{t}}$, has generator ${\textstyle {1 \over {2}}\Delta }$, where ${\displaystyle \Delta }$ denotes the Laplace operator.
• The two-dimensional process ${\displaystyle Y}$ satisfying:
${\displaystyle \mathrm {d} Y_{t}={\mathrm {d} t \choose \mathrm {d} B_{t}}}$
where ${\displaystyle B}$ is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator:
${\displaystyle {\mathcal {A}}f(t,x)={\frac {\partial f}{\partial t}}(t,x)+{\frac {1}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x)}$
• The Ornstein–Uhlenbeck process on ${\displaystyle \mathbb {R} }$, which satisfies the stochastic differential equation ${\textstyle dX_{t}=\theta (\mu -X_{t})dt+\sigma dB_{t}}$, has generator:
${\displaystyle {\mathcal {A}}f(x)=\theta (\mu -x)f'(x)+{\frac {\sigma ^{2}}{2}}f''(x)}$
• Similarly, the graph of the Ornstein–Uhlenbeck process has generator:
${\displaystyle {\mathcal {A}}f(t,x)={\frac {\partial f}{\partial t}}(t,x)+\theta (\mu -x){\frac {\partial f}{\partial x}}(t,x)+{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x)}$
• A geometric Brownian motion on ${\displaystyle \mathbb {R} }$, which satisfies the stochastic differential equation ${\textstyle dX_{t}=rX_{t}dt+\alpha X_{t}dB_{t}}$, has generator:
${\displaystyle {\mathcal {A}}f(x)=rxf'(x)+{\frac {1}{2}}\alpha ^{2}x^{2}f''(x)}$