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 Itō diffusion satisfying a stochastic differential equation of the form

:$mathrm\{d\}\; X\_\{t\}\; =\; b(X\_\{t\})\; ,\; mathrm\{d\}\; t\; +\; sigma\; (X\_\{t\})\; ,\; mathrm\{d\}\; B\_\{t\},$

where "B" is an "m"-dimensional Brownian motion and "b" : 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

:$A\; f\; (x)\; =\; lim\_\{t\; downarrow\; 0\}\; frac\{mathbf\{E\}^\{x\}\; [f(X\_\{t\})]\; -\; f(x)\}\{t\}.$

The set of all functions "f" for which this limit exists at a point "x" is denoted "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

::$mathrm\{d\}\; Y\_\{t\}\; =\; \{\; mathrm\{d\}\; t\; choose\; mathrm\{d\}\; B\_\{t\}\; \}\; ,$

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

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

* The Ornstein-Uhlenbeck process on R, which satisfies the stochastic differential equation d"X"_"t" = "μX"_"t" d"t" + "σ" d"B"_"t", has generator

::$A\; f(x)\; =\; mu\; x\; f\text{'}(x)\; +\; frac\{sigma^\{2\{2\}\; f"(x).$

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

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

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

::$A\; f(x)\; =\; r\; x\; f\text{'}(x)\; +\; frac1\{2\}\; alpha^\{2\}\; x^\{2\}\; f"(x).$

See also

*Dynkin's formula

References

* cite book
last = Øksendal
first = Bernt K.
authorlink = Bernt Øksendal
title = Stochastic Differential Equations: An Introduction with Applications
edition = Sixth edition
publisher=Springer
location = Berlin
year = 2003
id = ISBN 3-540-04758-1 (See Section 7)