128.131.95.200: /* References */

2013-08-21T12:34:23Z

References

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	~~Hi there~~. Let ~~me start by introducing~~ the ~~author~~, ~~her title~~ is ~~Sophia Boon but she by no means truly liked that title~~. ~~I am truly fond~~ of to go to ~~karaoke but I~~'~~ve been using on new things lately~~. ~~I am an invoicing officer and I~~'~~ll be promoted soon~~. ~~My spouse~~ and ~~I live~~ in ~~Kentucky~~.<br><br>~~My web~~-~~site~~ - ~~authentic psychic readings~~ - [~~http:~~//~~findyourflirt~~.~~net~~/~~index~~.~~php?m~~=~~member_profile&p~~=~~profile&~~id=~~117823 findyourflirt~~.~~net~~],		{{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]]

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Hi there. Let me start by introducing the author, her title is Sophia Boon but she by no means truly liked that title. I am truly fond of to go to karaoke but I've been using on new things lately. I am an invoicing officer and I'll be promoted soon. My spouse and I live in Kentucky.<br><br>My web-site - authentic psychic readings - [http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 findyourflirt.net],

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