Definitions

# 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 Itō diffusion satisfying a stochastic differential equation of the form

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

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

$A f \left(x\right) = lim_\left\{t downarrow 0\right\} frac\left\{mathbf\left\{E\right\}^\left\{x\right\} \left[f\left(X_\left\{t\right\}\right)\right] - f\left(x\right)\right\}\left\{t\right\}.$

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

$A f \left(x\right) = sum_\left\{i\right\} b_\left\{i\right\} \left(x\right) frac\left\{partial f\right\}\left\{partial x_\left\{i\right\}\right\} \left(x\right) + frac1\left\{2\right\} sum_\left\{i, j\right\} big\left(sigma \left(x\right) sigma \left(x\right)^\left\{top\right\} big\right)_\left\{i, j\right\} frac\left\{partial^\left\{2\right\} f\right\}\left\{partial x_\left\{i\right\} , partial x_\left\{j\right\}\right\} \left(x\right),$

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

$A f \left(x\right) = b\left(x\right) cdot nabla_\left\{x\right\} f\left(x\right) + frac1\left\{2\right\} big\left(sigma\left(x\right) sigma\left(x\right)^\left\{top\right\} big\right) : nabla_\left\{x\right\} nabla_\left\{x\right\} f\left(x\right).$

## 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

$mathrm\left\{d\right\} Y_\left\{t\right\} = \left\{ mathrm\left\{d\right\} t choose mathrm\left\{d\right\} B_\left\{t\right\} \right\} ,$

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

$A f\left(t, x\right) = frac\left\{partial f\right\}\left\{partial t\right\} \left(t, x\right) + frac1\left\{2\right\} frac\left\{partial^\left\{2\right\} f\right\}\left\{partial x^\left\{2\right\}\right\} \left(t, x\right).$

• The Ornstein-Uhlenbeck process on R, which satisfies the stochastic differential equation dXt = μXt dt + σ dBt, has generator

$A f\left(x\right) = mu x f\text{'}\left(x\right) + frac\left\{sigma^\left\{2\right\}\right\}\left\{2\right\} f\left(x\right).$

• Similarly, the graph of the Ornstein-Uhlenbeck process has generator

$A f\left(t, x\right) = frac\left\{partial f\right\}\left\{partial t\right\} \left(t, x\right) + mu x frac\left\{partial f\right\}\left\{partial x\right\} \left(t, x\right) + frac\left\{sigma^\left\{2\right\}\right\}\left\{2\right\} frac\left\{partial^\left\{2\right\} f\right\}\left\{partial x^\left\{2\right\}\right\} \left(t, x\right).$

• A geometric Brownian motion on R, which satisfies the stochastic differential equation dXt = rXt dt + αXt dBt, has generator

$A f\left(x\right) = r x f\text{'}\left(x\right) + frac1\left\{2\right\} alpha^\left\{2\right\} x^\left\{2\right\} f\left(x\right).$