# 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 $0$ 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. 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". 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". University of Sheffield. pp. 37–53.
8. ^ a b c Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591.
9. ^ Ito, K. (2006). Essentials of Stochastic Processes. American Mathematical Society. p. 54. ISBN 9780821838983.