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).

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.

Definition

General case

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

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

whenever this limit exists in ${\displaystyle {\overline {C_{c}^{0}(\mathbb {R} ^{d})}}\subset (C^{0}(\mathbb {R} ^{d}),\|\cdot \|_{\infty })}$, i.e. in the space of continuous functions ${\displaystyle \mathbb {R} ^{d}\to \mathbb {R} }$ vanishing at infinity.

This definition parallels the one of infinitesimal generator of ${\displaystyle C^{0}}$-semigroup.

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:

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

See also

• Dynkin's formula

References

• Calin, Ovidiu (2015). An Informal Introduction to Stochastic Calculus with Applications. Singapore: World Scientific Publishing. p. 315. ISBN 978-981-4678-93-3. (See Chapter 9)
• 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)