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Infinitesimal generator (stochastic processes)

From Wikipedia, the free encyclopedia

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).[citation needed]

Definition

General case

For a Feller process we define the generator by

whenever this limit exists in .

[more information needed]

Stochastic differential equations driven by Brownian motion

Let defined on a probability space be an Itô diffusion satisfying a stochastic differential equation of the form:

where is an m-dimensional Brownian motion and and are the drift and diffusion fields respectively. For a point , let denote the law of given initial datum , and let denote expectation with respect to .

The infinitesimal generator of is the operator , which is defined to act on suitable functions by:

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

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

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 , which satisfies the stochastic differential equation , has generator , where denotes the Laplace operator.
  • The two-dimensional process satisfying:
where 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 , which satisfies the stochastic differential equation , has generator:
  • Similarly, the graph of the Ornstein–Uhlenbeck process has generator:
  • A geometric Brownian motion on , which satisfies the stochastic differential equation , has generator:

See also

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)
This page was last edited on 13 February 2020, at 23:21
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