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{{About|infinitesimal generator for general stochastic processes|generators for continuous time Markov chains|transition rate matrix}} | |||
In [[mathematics]] — specifically, in [[stochastic processes|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 [[Lp space|''L''<sup>2</sup>]] [[Hermitian adjoint]] is used in evolution equations such as the [[Fokker–Planck equation]] (which describes the evolution of the [[probability density function]]s of the process). | |||
==Definition== | |||
Let ''X'' : [0, +∞) × Ω → '''R'''<sup>''n''</sup> defined on a [[probability space]] (Ω, Σ, '''P''') be an [[Itô diffusion]] satisfying a [[stochastic differential equation]] of the form | |||
:<math>\mathrm{d} X_{t} = b(X_{t}) \, \mathrm{d} t + \sigma (X_{t}) \, \mathrm{d} B_{t},</math> | |||
where ''B'' is an ''m''-dimensional [[Brownian motion]] and ''b'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup> and ''σ'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''×''m''</sup> are the drift and diffusion fields respectively. For a point ''x'' ∈ '''R'''<sup>''n''</sup>, let '''P'''<sup>''x''</sup> denote the law of ''X'' given initial datum ''X''<sub>0</sub> = ''x'', and let '''E'''<sup>''x''</sup> denote expectation with respect to '''P'''<sup>''x''</sup>. | |||
The '''infinitesimal generator''' of ''X'' is the operator ''A'', which is defined to act on suitable functions ''f'' : '''R'''<sup>''n''</sup> → '''R''' by | |||
:<math>A f (x) = \lim_{t \downarrow 0} \frac{\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}.</math> | |||
The set of all functions ''f'' for which this limit exists at a point ''x'' is denoted ''D''<sub>''A''</sub>(''x''), while ''D''<sub>''A''</sub> denotes the set of all ''f'' for which the limit exists for all ''x'' ∈ '''R'''<sup>''n''</sup>. One can show that any [[compact support|compactly-supported]] ''C''<sup>2</sup> (twice [[differentiable function|differentiable]] with [[continuous function|continuous]] second derivative) function ''f'' lies in ''D''<sub>''A''</sub> and that | |||
:<math>A f (x) = \sum_{i} b_{i} (x) \frac{\partial f}{\partial x_{i}} (x) + \frac1{2} \sum_{i, j} \big( \sigma (x) \sigma (x)^{\top} \big)_{i, j} \frac{\partial^{2} f}{\partial x_{i} \, \partial x_{j}} (x),</math> | |||
or, in terms of the [[gradient]] and [[dot product|scalar]] and [[Frobenius inner product|Frobenius]] [[inner product]]s, | |||
:<math>A f (x) = b(x) \cdot \nabla_{x} f(x) + \frac1{2} \big( \sigma(x) \sigma(x)^{\top} \big) : \nabla_{x} \nabla_{x} f(x).</math> | |||
==Generators of some common processes== | |||
* Standard Brownian motion on '''R'''<sup>''n''</sup>, which satisfies the stochastic differential equation d''X''<sub>''t''</sub> = d''B''<sub>''t''</sub>, has generator ½Δ, where Δ denotes the [[Laplace operator]]. | |||
* The two-dimensional process ''Y'' satisfying | |||
::<math>\mathrm{d} Y_{t} = { \mathrm{d} t \choose \mathrm{d} B_{t} } ,</math> | |||
: where ''B'' is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator | |||
::<math>A f(t, x) = \frac{\partial f}{\partial t} (t, x) + \frac1{2} \frac{\partial^{2} f}{\partial x^{2}} (t, x).</math> | |||
* The [[Ornstein–Uhlenbeck process]] on '''R''', which satisfies the stochastic differential equation d''X''<sub>''t''</sub> = ''θ'' (''μ'' − ''X''<sub>''t''</sub>) d''t'' + ''σ'' d''B''<sub>''t''</sub>, has generator | |||
::<math>A f(x) = \theta(\mu - x) f'(x) + \frac{\sigma^{2}}{2} f''(x).</math> | |||
* Similarly, the graph of the Ornstein–Uhlenbeck process has generator | |||
::<math>A f(t, x) = \frac{\partial f}{\partial t} (t, x) + \theta(\mu - x) \frac{\partial f}{\partial x} (t, x) + \frac{\sigma^{2}}{2} \frac{\partial^{2} f}{\partial x^{2}} (t, x).</math> | |||
* A [[geometric Brownian motion]] on '''R''', which satisfies the stochastic differential equation d''X''<sub>''t''</sub> = ''rX''<sub>''t''</sub> d''t'' + ''αX''<sub>''t''</sub> d''B''<sub>''t''</sub>, has generator | |||
::<math>A f(x) = r x f'(x) + \frac1{2} \alpha^{2} x^{2} f''(x).</math> | |||
== 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 | |||
| publisher=Springer | |||
| location = Berlin | |||
| year = 2003 | |||
| id = ISBN 3-540-04758-1 | |||
| doi = 10.1007/978-3-642-14394-6 | |||
}} (See Section 7.3) | |||
[[Category:Stochastic differential equations]] | |||
Latest revision as of 13:34, 21 August 2013
29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. 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 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).
Definition
Let X : [0, +∞) × Ω → Rn 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 : Rn → Rn and σ : Rn → Rn×m are the drift and diffusion fields respectively. For a point x ∈ Rn, let Px denote the law of X given initial datum X0 = x, and let Ex denote expectation with respect to Px.
The infinitesimal generator of X is the operator A, which is defined to act on suitable functions f : Rn → R by
The set of all functions f for which this limit exists at a point x is denoted DA(x), while DA denotes the set of all f for which the limit exists for all x ∈ Rn. One can show that any compactly-supported C2 (twice differentiable with continuous second derivative) function f lies in DA and that
or, in terms of the gradient and scalar and Frobenius inner products,
Generators of some common processes
- Standard Brownian motion on Rn, which satisfies the stochastic differential equation dXt = dBt, 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 dXt = θ (μ − Xt) dt + σ dBt, has generator
- Similarly, the graph of the Ornstein–Uhlenbeck process has generator
- A geometric Brownian motion on R, which satisfies the stochastic differential equation dXt = rXt dt + αXt dBt, has generator
See also
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (See Section 7.3)