Cauchy process

百度　　5.苦瓜　　一般来说,苦味食品都具有解读功能,苦瓜是日常经常会食用的蔬菜,凉拌或清炒皆可。

In probability theory, a Cauchy process is a type of stochastic process. 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:

Symmetric Cauchy process

The symmetric Cauchy process can be described by a Brownian motion or Wiener process subject to a Lévy subordinator.^[7] The Lévy subordinator is a process associated with a Lévy distribution having location parameter of $0$ and a scale parameter of $t^{2}/2$ .^[7] The Lévy distribution is a special case of the inverse-gamma distribution. So, using $C$ to represent the Cauchy process and $L$ to represent the Lévy subordinator, the symmetric Cauchy process can be described as:

C(t;0,1)\;:=\;W(L(t;0,t^{2}/2)).

The Lévy 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 Lévy–Khintchine representation for the symmetric Cauchy process is a triplet with zero drift and zero diffusion, giving a Lévy–Khintchine triplet of $(0,0,W)$ , where $W(dx)=dx/(\pi x^{2})$ .^[8]

The marginal characteristic function of the symmetric Cauchy process has the form:^[1]^[8]

\operatorname {E} {\Big [}e^{i\theta 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 \over \pi }\left[{t \over x^{2}+t^{2}}\right].

Asymmetric Cauchy process

The asymmetric Cauchy process is defined in terms of a parameter $\beta$ . Here $\beta$ is the skewness parameter, and its absolute value must be less than or equal to 1.^[1] In the case where $|\beta |=1$ the process is considered a completely asymmetric Cauchy process.^[1]

The Lévy–Khintchine triplet has the form $(0,0,W)$ , where $W(dx)={\begin{cases}Ax^{-2}\,dx&{\text{if }}x>0\\Bx^{-2}\,dx&{\text{if }}x<0\end{cases}}$ , where $A\neq B$ , $A>0$ and $B>0$ .^[1]

Given this, $\beta$ is a function of $A$ and $B$ .

The characteristic function of the asymmetric Cauchy distribution has the form:^[1]

\operatorname {E} {\Big [}e^{i\theta X_{t}}{\Big ]}=e^{-t(|\theta |+i\beta \theta \ln |\theta |/(2\pi ))}.

The marginal probability distribution of the asymmetric Cauchy process is a stable distribution with index of stability (i.e., α parameter) equal to 1.

References

^ ^a ^b ^c ^d ^e ^f ^g Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701.
^ ^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. 228. ISBN 9783540307884.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Winkel, M. "Introduction to Levy processes" (PDF). pp. 15–16. Retrieved 2025-08-07.
^ Jacob, N. (2005). Pseudo Differential Operators & Markov Processes: Markov Processes And Applications, Volume 3. Imperial College Press. p. 135. ISBN 9781860945687.
^ Bertoin, J. (2001). "Some elements on Lévy processes". In Shanbhag, D.N. (ed.). Stochastic Processes: Theory and Methods. Gulf Professional Publishing. p. 122. ISBN 9780444500144.
^ Kroese, D.P.; Taimre, T.; Botev, Z.I. (2011). Handbook of Monte Carlo Methods. John Wiley & Sons. p. 214. ISBN 9781118014950.
^ ^a ^b ^c Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.
^ ^a ^b ^c Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591.
^ It?, K. (2006). Essentials of Stochastic Processes. American Mathematical Society. p. 54. ISBN 9780821838983.

[models-1] ^ ^a ^b ^c ^d ^e ^f ^g Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701.

[stable-2] 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. 228. ISBN 9783540307884.{{cite book}}: CS1 maint: multiple names: authors list (link)

[Winkel-3] Winkel, M. "Introduction to Levy processes" (PDF). pp. 15–16. Retrieved 2025-08-07.

[4] Jacob, N. (2005). Pseudo Differential Operators & Markov Processes: Markov Processes And Applications, Volume 3. Imperial College Press. p. 135. ISBN 9781860945687.

[5] Bertoin, J. (2001). "Some elements on Lévy processes". In Shanbhag, D.N. (ed.). Stochastic Processes: Theory and Methods. Gulf Professional Publishing. p. 122. ISBN 9780444500144.

[6] Kroese, D.P.; Taimre, T.; Botev, Z.I. (2011). Handbook of Monte Carlo Methods. John Wiley & Sons. p. 214. ISBN 9781118014950.

[applebaum-7] Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.

[cinlar-8] Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591.

[essentials-9] It?, K. (2006). Essentials of Stochastic Processes. American Mathematical Society. p. 54. ISBN 9780821838983.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

v t e Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Additive process Airy process Bessel process Birth–death process pure birth Brownian motion Bridge Dyson Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems It? diffusion It? process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Quasimartingale Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Determinantal Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White Korn-Kreer-Lenssen LIBOR market Rendleman–Bartter SABR volatility Va?í?ek Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise-deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator It? integral It?'s lemma Karhunen–Loève theorem Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
List of topics Category

石榴什么时候成熟	私密是什么意思	什么是碱性磷酸酶高怎么回事	拔智齿挂什么科	脂肪肝适合吃什么水果
喉结下面是什么部位	卵巢增大是什么原因引起的	苁蓉有什么功效	什么牌子的助听器最好	乳腺结节有什么症状
什么样的升旗仪式	什么是内分泌失调	脚突然肿了是什么原因	蚧壳虫用什么药最有效	周杰伦英文名叫什么
为什么头老是晕晕的	刘字是什么偏旁	1994年五行属什么	一月十八号是什么星座	儿童喝蜂蜜水有什么好处和坏处

冠心病吃什么药最好hcv9jop0ns1r.cn	策字五行属什么hcv9jop2ns4r.cn	志愿号是什么意思hcv7jop9ns5r.cn	2000年属什么的hcv8jop3ns1r.cn	尿酸高能吃什么hcv9jop1ns3r.cn
再生纤维是什么面料hcv8jop5ns7r.cn	惊奇地什么hcv8jop9ns8r.cn	尿道感染有什么现象clwhiglsz.com	尿酸高的人吃什么食物好hcv8jop1ns8r.cn	diff什么意思hcv9jop6ns9r.cn
神母是什么病hcv8jop2ns9r.cn	拉屎黑色的是什么原因hcv7jop9ns9r.cn	什么时候可以考研hcv9jop7ns5r.cn	黄芪什么味道creativexi.com	西瓜和什么食物相克hcv8jop2ns5r.cn
摇篮是什么意思hcv7jop7ns1r.cn	11月20号是什么星座hcv9jop3ns6r.cn	桑螵蛸是什么hcv9jop2ns9r.cn	孕前检查挂什么科hcv9jop0ns3r.cn	栀子花什么时候修剪creativexi.com