# 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:

1. It is a Lévy process[3][4][5]
2. It is a stable process[1][2]
3. It is a pure jump process[6]
4. 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 ${\displaystyle 0}$ and a scale parameter of ${\displaystyle t^{2}/2}$.[7] The Lévy distribution is a special case of the inverse-gamma distribution. So, using ${\displaystyle C}$ to represent the Cauchy process and ${\displaystyle L}$ to represent the Lévy subordinator, the symmetric Cauchy process can be described as:

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

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 ${\displaystyle (0,0,W)}$, where ${\displaystyle W(dx)=dx/(\pi x^{2})}$.[8]

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

${\displaystyle \operatorname {E} {\Big [}e^{i\theta X_{t}}{\Big ]}=e^{-t|\theta |}.}$

The marginal probability distribution of the symmetric Cauchy process is the Cauchy distribution whose density is[8][9]

${\displaystyle f(x;t)={1 \over \pi }\left[{t \over x^{2}+t^{2}}\right].}$

## Asymmetric Cauchy process

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

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

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

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

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

The marginal probability distribution of the asymmetric Cauchy process is a stable distribution with index of stability equal to 1.

## References

1. Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701.
2. ^ 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. From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift. Springer. p. 228. ISBN 9783540307884.
3. ^ Winkel, M. "Introduction to Levy processes" (PDF). pp. 15–16. Retrieved 2013-02-07.
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. 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.
7. ^ a b c Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.
8. ^ a b c Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591.
9. ^ Itô, K. (2006). Essentials of Stochastic Processes. American Mathematical Society. p. 54. ISBN 9780821838983.