Infinitesimal generator (stochastic processes)

This article is about infinitesimal generator for general stochastic processes. For generators for the special case of finite-state continuous time Markov chains, see transition rate matrix.

In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a stochastic process 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|date=January 2020|reason=There is no citation explaining the general case of an infinitesimal generator}

Definition

General case

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

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

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

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:

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

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

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

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

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

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

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

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

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 $\mathbb {R} ^{n}$, which satisfies the stochastic differential equation $dX_{t}=dB_{t}$, has generator ${\textstyle {1 \over {2}}\Delta }$, where $\Delta$ denotes the Laplace operator.
• The two-dimensional process $Y$ satisfying:

$\mathrm {d} Y_{t}={\mathrm {d} t \choose \mathrm {d} B_{t}}$

where $B$ is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator:

${\mathcal {A}}f(t,x)={\frac {\partial f}{\partial t}}(t,x)+{\frac {1}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x)$

• The Ornstein–Uhlenbeck process on $\mathbb {R}$, which satisfies the stochastic differential equation ${\textstyle dX_{t}=\theta (\mu -X_{t})dt+\sigma dB_{t}}$, has generator:

${\mathcal {A}}f(x)=\theta (\mu -x)f'(x)+{\frac {\sigma ^{2}}{2}}f''(x)$

• Similarly, the graph of the Ornstein–Uhlenbeck process has generator:

${\mathcal {A}}f(t,x)={\frac {\partial f}{\partial t}}(t,x)+\theta (\mu -x){\frac {\partial f}{\partial x}}(t,x)+{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x)$

• A geometric Brownian motion on $\mathbb {R}$, which satisfies the stochastic differential equation ${\textstyle dX_{t}=rX_{t}dt+\alpha X_{t}dB_{t}}$, has generator:

${\mathcal {A}}f(x)=rxf'(x)+{\frac {1}{2}}\alpha ^{2}x^{2}f''(x)$

See also

• Dynkin's formula

References

• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. doi:10.1007/978-3-642-14394-6. ISBN 3-540-04758-1. (See Section 7.3)