# 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 *L*^{2} 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 → **R**^{n} defined on a probability space (Ω, Σ, **P**) be an Itô diffusion satisfying a stochastic differential equation of the form

where *B* is an *m*-dimensional Brownian motion and *b* : **R**^{n} → **R**^{n} and *σ* : **R**^{n} → **R**^{n×m} are the drift and diffusion fields respectively. For a point *x* ∈ **R**^{n}, let **P**^{x} denote the law of *X* given initial datum *X*_{0} = *x*, and let **E**^{x} denote expectation with respect to **P**^{x}.

The **infinitesimal generator** of *X* is the operator *A*, which is defined to act on suitable functions *f* : **R**^{n} → **R** by

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

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

## Generators of some common processes

- Standard Brownian motion on
**R**^{n}, which satisfies the stochastic differential equation d*X*_{t}= d*B*_{t}, has generator ½Δ, where Δ denotes the Laplace operator. - The two-dimensional process
*Y*satisfying

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

- The Ornstein–Uhlenbeck process on
**R**, which satisfies the stochastic differential equation d*X*_{t}=*θ*(*μ*−*X*_{t}) d*t*+*σ*d*B*_{t}, has generator

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

- A geometric Brownian motion on
**R**, which satisfies the stochastic differential equation d*X*_{t}=*rX*_{t}d*t*+*αX*_{t}d*B*_{t}, has generator

## See also

## References

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