[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/cauchy-process-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/cauchy-process-wikipedia\/","headline":"Cauchy process – Wikipedia","name":"Cauchy process – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 In probability theory, a Cauchy process is a type of stochastic process. 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There are symmetric and asymmetric forms of the Cauchy process.[1]   The unspecified  term “Cauchy process” is often used to refer to the symmetric Cauchy process.[2]The Cauchy process has a number of properties:       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4It is a L\u00e9vy process[3][4][5]It is a stable process[1][2]It is a pure jump process[6]Its moments are infinite.Symmetric Cauchy process[edit]The symmetric Cauchy process can be described by a Brownian motion or Wiener process subject to a L\u00e9vy subordinator.[7]  The L\u00e9vy subordinator is a process associated with a L\u00e9vy distribution having location parameter of 0{displaystyle 0} and a scale parameter of t2\/2{displaystyle t^{2}\/2}.[7]  The L\u00e9vy distribution is a special case of the inverse-gamma distribution.  So, using C{displaystyle C} to represent the Cauchy process and        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4L{displaystyle L} to represent the L\u00e9vy subordinator, the symmetric Cauchy process can be described as:C(t;0,1):=W(L(t;0,t2\/2)).{displaystyle C(t;0,1);:=;W(L(t;0,t^{2}\/2)).}The L\u00e9vy distribution is the probability of the first hitting time for a Brownian motion, and thus the Cauchy process is essentially the result of two independent Brownian motion processes.[7]The  L\u00e9vy\u2013Khintchine representation for the symmetric Cauchy process is a triplet with zero drift and zero diffusion, giving a L\u00e9vy\u2013Khintchine triplet of (0,0,W){displaystyle (0,0,W)}, where W(dx)=dx\/(\u03c0x2){displaystyle W(dx)=dx\/(pi x^{2})}.[8]The marginal characteristic function of the symmetric Cauchy process has the form:[1][8]E\u2061[ei\u03b8Xt]=e\u2212t|\u03b8|.{displaystyle operatorname {E} {Big [}e^{itheta X_{t}}{Big ]}=e^{-t|theta |}.}The marginal probability distribution of the  symmetric Cauchy process is the Cauchy distribution whose density is[8][9]f(x;t)=1\u03c0[tx2+t2].{displaystyle f(x;t)={1 over pi }left[{t over x^{2}+t^{2}}right].}Asymmetric Cauchy process[edit]The asymmetric Cauchy process is defined in terms of a parameter \u03b2{displaystyle beta }. Here\u03b2{displaystyle beta } is the skewness parameter, and its absolute value must be less than or equal to 1.[1]  In the case where |\u03b2|=1{displaystyle |beta |=1} the process is considered a completely asymmetric Cauchy process.[1]The L\u00e9vy\u2013Khintchine triplet has the form (0,0,W){displaystyle (0,0,W)}, where W(dx)={Ax\u22122dxif\u00a0x>0Bx\u22122dxif\u00a0x0\\Bx^{-2},dx&{text{if }}x, where A\u2260B{displaystyle Aneq B}, 0″\/> and 0″\/>.[1]Given this, \u03b2{displaystyle beta } is a function of A{displaystyle A} and B{displaystyle B}.The characteristic function of the asymmetric Cauchy distribution has the form:[1]E\u2061[ei\u03b8Xt]=e\u2212t(|\u03b8|+i\u03b2\u03b8ln\u2061|\u03b8|\/(2\u03c0)).{displaystyle operatorname {E} {Big [}e^{itheta X_{t}}{Big ]}=e^{-t(|theta |+ibeta theta ln |theta |\/(2pi ))}.}The marginal probability distribution of the asymmetric Cauchy process is a stable distribution with index of stability (i.e., \u03b1 parameter) equal to 1.References[edit]^ a b c d e f g Kovalenko, I.N.;  et\u00a0al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp.\u00a0210\u2013211. ISBN\u00a09780849328701.^ a b Engelbert, H.J., Kurenok, V.P. & Zalinescu, A. (2006). “On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes”.  In Kabanov, Y.; Liptser, R.; Stoyanov, J. (eds.). From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift. Springer. p.\u00a0228. ISBN\u00a09783540307884.{{cite book}}:  CS1 maint: multiple names: authors list (link)^ Winkel, M. “Introduction to Levy processes” (PDF). pp.\u00a015\u201316. Retrieved 2013-02-07.^ Jacob, N. (2005). Pseudo Differential Operators & Markov Processes: Markov Processes And Applications, Volume 3. Imperial College Press. p.\u00a0135. ISBN\u00a09781860945687.^ Bertoin, J. (2001). “Some elements on L\u00e9vy processes”.  In Shanbhag, D.N. (ed.). Stochastic Processes: Theory and Methods. Gulf Professional Publishing. p.\u00a0122. ISBN\u00a09780444500144.^ Kroese, D.P.; Taimre, T.; Botev, Z.I. (2011). Handbook of Monte Carlo Methods. John Wiley & Sons. p.\u00a0214. ISBN\u00a09781118014950.^ a b c Applebaum, D. “Lectures on L\u00e9vy processes and Stochastic calculus, Braunschweig; Lecture 2: L\u00e9vy processes” (PDF). University of Sheffield. pp.\u00a037\u201353.^ a b c Cinlar, E. (2011). Probability and Stochastics. Springer. p.\u00a0332. ISBN\u00a09780387878591.^ It\u00f4, K. (2006). Essentials of Stochastic Processes. American Mathematical Society. p.\u00a054. ISBN\u00a09780821838983.       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