Cauchy process explained

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)} .

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

\operatorname{E} [ e ^{i\theta X _{t}} ] = e ^{-t} .

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 & if x> 0 \ Bx ^{-2} dx & 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} [ e ^{i\theta X _{t}} ] = e ^{-t} .

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

Notes and References

1. Book: Models of Random Processes: A Handbook for Mathematicians and Engineers. 210–211. Kovalenko, I.N.. 1996. CRC Press. 9780849328701. etal.
2. Book: From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift. limited. Kabanov, Y. . Liptser, R. . Stoyanov, J. . On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes. Engelbert, H.J., Kurenok, V.P. & Zalinescu, A.. 228. 2006. Springer. 9783540307884.
3. Web site: Introduction to Levy processes. Winkel, M.. 15–16. 2013-02-07.
4. Book: Pseudo Differential Operators & Markov Processes: Markov Processes And Applications, Volume 3. Jacob, N.. 135. 2005. Imperial College Press. 9781860945687.
5. Book: Stochastic Processes: Theory and Methods. Shanbhag, D.N.. Some elements on Lévy processes. Bertoin, J.. 122. 2001. Gulf Professional Publishing. 9780444500144.
6. Book: Handbook of Monte Carlo Methods. limited. Kroese, D.P. . Dirk Kroese . Taimre, T. . Botev, Z.I. . 214. 2011. John Wiley & Sons. 9781118014950.
7. Web site: Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes. Applebaum, D.. 37–53. University of Sheffield.
8. Book: Probability and Stochastics. limited. Cinlar, E.. 332. 2011. Springer. 9780387878591.
9. Book: Essentials of Stochastic Processes. Itô, K.. 54. American Mathematical Society. 2006. 9780821838983.