Cauchy process

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:

It is a Lévy 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

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^[9]^[8]

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 $A \ne B$ , $A>0$ 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 equal to 1.

References

↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Kovalenko, I.N. et al (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701.
↑ 2.0 2.1 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. From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift. Springer. p. 228. ISBN 9783540307884.
↑ Winkel, M. "Introduction to Levy processes". pp. 15–16. Retrieved 2013-02-07.
↑ 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. 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.
↑ 7.0 7.1 7.2 Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes". University of Sheffield. pp. 37–53.
↑ 8.0 8.1 8.2 Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591.
↑ Ito, K. (2006). Essentials of Stochastic Processes. American Mathematical Society. p. 54. ISBN 9780821838983.

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

Continuous time	Bessel process Birth–death process Brownian motion Bridge 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 Hunt process Interacting particle systems Itō diffusion Itō process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process 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 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 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	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market SABR volatility Vašíček

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

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

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 Itō integral Itō's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prohorov 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 Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

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