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 (i.e., α parameter) 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. 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.CS1 maint: multiple names: authors list (link)
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. (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.
7. Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.
8. 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.