# Infinitesimal generator (stochastic processes)

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).

## Definition

Let X : [0, +∞) × Ω → Rn defined on a probability space (Ω, Σ, 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 B is an m-dimensional Brownian motion and b : Rn → Rn and σ : Rn → Rn×m are the drift and diffusion fields respectively. For a point x ∈ Rn, let Px denote the law of X given initial datum X0 = x, and let Ex denote expectation with respect to Px.

The infinitesimal generator of X is the operator A, which is defined to act on suitable functions f : Rn → R by

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

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

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

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

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

## Generators of some common processes

• Standard Brownian motion on Rn, which satisfies the stochastic differential equation dXt = dBt, has generator ½Δ, where Δ denotes the Laplace operator.
• The two-dimensional process Y satisfying
${\displaystyle \mathrm {d} Y_{t}={\mathrm {d} t \choose \mathrm {d} B_{t}},}$
where B is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator
${\displaystyle Af(t,x)={\frac {\partial f}{\partial t}}(t,x)+{\frac {1}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x).}$
• The Ornstein–Uhlenbeck process on R, which satisfies the stochastic differential equation dXt = θ (μ − Xt) dt + σ dBt, has generator
${\displaystyle Af(x)=\theta (\mu -x)f'(x)+{\frac {\sigma ^{2}}{2}}f''(x).}$
• Similarly, the graph of the Ornstein–Uhlenbeck process has generator
${\displaystyle Af(t,x)={\frac {\partial f}{\partial t}}(t,x)+\theta (\mu -x){\frac {\partial f}{\partial x}}(t,x)+{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x).}$
• A geometric Brownian motion on R, which satisfies the stochastic differential equation dXt = rXt dt + αXt dBt, has generator
${\displaystyle Af(x)=rxf'(x)+{\frac {1}{2}}\alpha ^{2}x^{2}f''(x).}$