Infinitesimal generator (stochastic processes)

Infinitesimal generator (stochastic processes)

In mathematics &mdash; specifically, in stochastic analysis &mdash; 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, +&infin;) &times; &Omega; &rarr; R"n" defined on a probability space (&Omega;, &Sigma;, 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" : R"n" &rarr; R"n" and "&sigma;" : R"n" &rarr; R"n"&times;"m" are the drift and diffusion fields respectively. For a point "x" &isin; 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" &rarr; 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 "D""A"("x"), while "D""A" denotes the set of all "f" for which the limit exists for all "x" &isin; 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 ½&Delta;, where &Delta; 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

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

* The Ornstein-Uhlenbeck process on R, which satisfies the stochastic differential equation d"X""t" = "&mu;X""t" d"t" + "&sigma;" d"B""t", has generator

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

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

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

* 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

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

*Dynkin's formula

References

* cite book
last = Øksendal
first = Bernt K.
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)

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