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| [[File:Local times surface.png|thumb|300px|A sample path of an Itō process together with its surface of local times.]]
| | The individual who wrote the post is called Jayson Hirano and he completely digs that name. For years she's been operating as a journey agent. Mississippi is exactly where his house is. The preferred hobby for him and his children is fashion and he'll be beginning some thing else alongside with it.<br><br>My site: psychics online; [http://appin.co.kr/board_Zqtv22/688025 appin.co.kr], |
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| In the [[mathematics|mathematical]] theory of [[stochastic process]]es, '''local time''' is a stochastic process associated with [[diffusion]] processes such as [[Brownian motion]], that characterizes the amount of time a particle has spent at a given level. Local time appears in various [[stochastic integration]] formulas, such as [[Tanaka's formula]], if the integrand is not sufficiently smooth. It is also studied in statistical mechanics in the context of [[random field]]s.
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| ==Formal definition==
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| For a diffusion process <math>(b_s)_{s\ge 0}</math>, the local time of <math>b</math> at the point <math>x</math> is the stochastic process
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| :<math>L^x(t) =\int_0^t \delta(x-b(s))\,ds,</math>
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| where <math>\delta</math> is the [[Dirac delta function]]. It is a notion invented by [[Paul Lévy (mathematician)|Paul Lévy]]. The basic idea is that <math>L^x(t)</math> is a (rescaled) measure of how much time <math>b(s)</math> has spent at <math>x</math> up to time <math>t</math>. It may be written as
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| :<math> L^x(t) =\lim_{\varepsilon\downarrow 0} \frac{1}{2\varepsilon} \int_0^t 1_{\{ x- \varepsilon < b(s) < x+\varepsilon \}} \, ds,</math>
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| which explains why it is called the local time of <math>b</math> at <math>x</math>. For a discrete state-space process <math>(X_s)_{s\ge 0}</math>, the local time can be expressed more simply as<ref>{{cite book |first=Ioannis |last=Karatas |first2=Steven |last2=Shreve |year=1991 |title=Brownian Motion and Stochastic Calculus |publisher=Springer }}</ref>
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| :<math> L^x(t) =\int_0^t 1_{\{x\}}(X_s) \, ds.</math>
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| ==Tanaka's Formula==
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| Tanaka's formula provides a definition of local time for an arbitrary continuous semimartingale <math>(X_s)_{s\ge 0}</math> on <math> \mathbb R: </math><ref name="Kallenberg">{{cite book |last=Kallenberg |title=Foundations of Modern Probability |location=New York |publisher=Springer |year=1997 |pages=428–449 |isbn=0387949577 }}</ref>
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| : <math> L^x(t) = |X_t - x| - |X_0 - x| - \int_{0}^t \left( 1_{(0,\infty)}(X_s - x) - 1_{(-\infty, 0]}(X_s-x) \right) dX_s, \qquad t \geq 0. </math>
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| A more general form was proven independently by Meyer<ref>{{cite book |first=P. A. |last=Meyer |chapter=Un cours sur les intégrales stochastiques |title=Séminaire de probabilités 1967–1980 |series=[[Lecture Notes in Mathematics|Lect. Notes in Math.]] |volume=1771 |issue= |pages=174–329 |year=2002 |origyear=1976 |doi=10.1007/978-3-540-45530-1_11 }}</ref> and Wang;<ref>{{cite journal |last=Wang |title=Generalized Itô's formula and additive functionals of Brownian motion |journal=Z. Wahrsch. verw. Geb. |volume=41 |issue= |pages=153–159 |year=1977 }}</ref> the formula extends Itô's lemma for twice differentiable functions to a more general class of functions. If <math> F:\mathbb R \rightarrow \mathbb R</math> is absolutely continuous with derivative <math> F',</math> which is of bounded variation, then
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| : <math> F(X_t) = F(X_0) + \int_{0}^t F'_{-}(X_s) dX_s + \frac12 \int_{-\infty}^\infty L^x(t) dF'(x), </math>
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| where <math> F'_{-}</math> is the left derivative. | |
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| Tanaka's formula can be used to show that the field of local times <math> L = (L^x(t))_{x \in \mathbb R, t \geq 0}</math> has a modification which is [[càdlàg]] in <math> x</math>, and uniformly bounded in <math> t</math>.<ref name="Kallenberg"/>
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| Tanaka's formula provides the explicit [[Doob-Meyer decomposition theorem|Doob-Meyer decomposition]] for the one-dimensional reflecting Brownian motion, <math>(|B_s|)_{s \geq 0}</math>.
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| ==Ray-Knight Theorems==
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| The field of local times <math> L_t = (L^x_t)_{x \in E}</math> associated to a stochastic process on a space <math>E</math> is a well studied topic in the area of random fields. Ray-Knight type theorems relate the field ''L''<sub>t</sub> to an associated [[Gaussian process]].
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| In general Ray-Knight type theorems of the first kind consider the field ''L''<sub>t</sub> at a hitting time of the underlying process, whilst theorems of the second time are in terms of a stopping time at which the field of local times first exceeds a given value.
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| ===First Ray-Knight Theorem===
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| Let (''B''<sub>t</sub>)<sub>t ≥ 0</sub> be a one-dimensional Brownian motion started from ''B''<sub>0</sub> = ''a'' > ''0'', and (''W''<sub>t</sub>)<sub>t≥0</sub> be a standard two-dimensional Brownian motion ''W''<sub>0</sub> = ''0'' ''∈'' '''R'''<sup>2</sup>. Define the stopping time at which ''B'' first hits the origin, <math> T = \inf\{t \geq 0 \colon B_t = 0\}</math>. Ray<ref>{{cite journal |first=D. |last=Ray |title=Sojourn times of a diffusion process |journal=[[Illinois Journal of Mathematics|Illinois J. Math.]] |volume=7 |issue=4 |pages=615–630 |year=1963 |doi= |mr=0156383 |zbl=0118.13403 }}</ref> and Knight<ref>{{cite journal |first=F. B. |last=Knight |title=Random walks and a sojourn density process of Brownian motion |journal=[[Transactions of the American Mathematical Society|Trans. Amer. Math. Soc.]] |volume=109 |issue=1 |pages=56–86 |year=1963 |jstor=1993647 }}</ref> (independently) showed that
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| {{NumBlk|:|<math>\left\{ L^x(T) \colon x \in [0,a] \right\} \stackrel{\mathcal{D}}{=} \left\{ |W_x|^2 \colon x \in [0,a] \right\} \,</math>|{{EquationRef|1}}}}
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| where (''L''<sub>t</sub>)<sub>t ≥ 0</sub> is the field of local times of (''B''<sub>t</sub>)<sub>t ≥ 0</sub>, and equality is in distribution on ''C''[''0'', ''a'']. The process |''W''<sub>x</sub>|<sup>2</sup> is known as the squared Bessel process.
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| ===Second Ray-Knight Theorem===
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| Let (''B''<sub>t</sub>)<sub>t ≥ 0</sub> be a standard one-dimensional Brownian motion ''B''<sub>0</sub> = ''0'' ''∈'' '''R''', and let (''L''<sub>t</sub>)<sub>t ≥ 0</sub> be the associated field of local times. Let ''T''<sub>a</sub> be the first time at which the local time at zero exceeds ''a'' > ''0''
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| : <math> T_a = \inf \{ t \geq 0 \colon L^0_t > a \}.</math>
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| Let (''W''<sub>t</sub>)<sub>t ≥ 0</sub> be an independent one-dimensional Brownian motion started from ''B''<sub>0</sub> = ''a'' > ''0'', then<ref>{{cite book |last=Marcus |last2=Rosen |title=Markov Processes, Gaussian Processes and Local Times |location=New York |publisher=Cambridge University Press |year=2006 |pages=53–56 |isbn=0521863007 }}</ref>
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| {{NumBlk|:|<math>\left \{ L^x_{T_a} + W_x^2 \colon x \geq 0 \right \} \stackrel{\mathcal{D}}{=} \left\{ (W_x + \sqrt a )^2 \colon x \geq 0 \right \}. \,</math>|{{EquationRef|2}}}}
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| Equivalently, the process <math>(L^x_{T_a})_{x \geq 0}</math> (which is a process in the spatial variable <math>x</math>) is equal in distribution to the squared Bessel process, and as such is Markovian.
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| ===Generalized Ray-Knight Theorems===
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| Results of Ray-Knight type for more general stochastic processes have been intensively studied, and analogue statements of both ({{EquationNote|1}}) and ({{EquationNote|2}}) are known for strongly symmetric Markov processes.
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| ==See also==
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| * [[Tanaka's formula]]
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| * [[Brownian motion]]
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| * [[Random field]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *K. L. Chung and R. J. Williams, ''Introduction to Stochastic Integration'', 2nd edition, 1990, Birkhäuser, ISBN 978-0-8176-3386-8.
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| * M. Marcus and J. Rosen, ''Markov Processes, Gaussian Processes, and Local Times'', 1st edition, 2006, Cambridge University Press ISBN 978-0-521-86300-1
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| *P.Morters and Y.Peres, ''Brownian Motion'', 1st edition, 2010, Cambridge University Press, ISBN 978-0-521-76018-8.
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| {{Stochastic processes}}
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| [[Category:Stochastic processes]]
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| [[Category:Statistical mechanics]]
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The individual who wrote the post is called Jayson Hirano and he completely digs that name. For years she's been operating as a journey agent. Mississippi is exactly where his house is. The preferred hobby for him and his children is fashion and he'll be beginning some thing else alongside with it.
My site: psychics online; appin.co.kr,