Infinitesimal generator (stochastic processes)

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

:A f (x) = sum_{i} b_{i} (x) frac{partial f}{partial x_{i (x) + frac1{2} sum_{i, j} ig( sigma (x) sigma (x)^{ op} ig)_{i, j} frac{partial^{2} f}{partial x_{i} , partial x_{j (x),

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

:A f (x) = b(x) cdot abla_{x} f(x) + frac1{2} ig( sigma(x) sigma(x)^{ op} ig) : abla_{x} abla_{x} f(x).

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'(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'(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)


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