Cauchy process

In probability theory, a Cauchy process is a type of stochastic process. There are symmetric and asymmetric forms of the Cauchy process.{{cite book|title=Models of Random Processes: A Handbook for Mathematicians and Engineers|pages=210–211|author=Kovalenko, I.N.|year=1996|publisher=CRC Press|isbn=9780849328701|display-authors=etal}} The unspecified term "Cauchy process" is often used to refer to the symmetric Cauchy process.{{cite book|title=From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift|url=https://archive.org/details/fromstochasticca00kaba_517|url-access=limited|editor1=Kabanov, Y. |editor2=Liptser, R. |editor3=Stoyanov, J. |chapter=On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes|author=Engelbert, H.J., Kurenok, V.P. & Zalinescu, A.|page=[https://archive.org/details/fromstochasticca00kaba_517/page/n261 228]|year=2006|publisher=Springer|isbn=9783540307884}}

The Cauchy process has a number of properties:

It is a Lévy process{{cite web|title=Introduction to Levy processes|author=Winkel, M.|pages=15–16|url=http://www.stats.ox.ac.uk/~winkel/lp1.pdf|access-date=2013-02-07}}{{cite book|title=Pseudo Differential Operators & Markov Processes: Markov Processes And Applications, Volume 3|author=Jacob, N.|page=135|year=2005|publisher=Imperial College Press|isbn=9781860945687}}{{cite book|title=Stochastic Processes: Theory and Methods|editor=Shanbhag, D.N.|chapter=Some elements on Lévy processes|author=Bertoin, J.|page=122|year=2001|publisher=Gulf Professional Publishing|isbn=9780444500144}}
It is a stable process
It is a pure jump process{{cite book|title=Handbook of Monte Carlo Methods|url=https://archive.org/details/handbookmontecar00kroe|url-access=limited|author1=Kroese, D.P. |author-link1=Dirk Kroese |author2=Taimre, T. |author3=Botev, Z.I. |page=[https://archive.org/details/handbookmontecar00kroe/page/n236 214]|year=2011|publisher=John Wiley & Sons|isbn=9781118014950}}
Its moments are infinite.

Symmetric Cauchy process

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The symmetric Cauchy process can be described by a Brownian motion or Wiener process subject to a Lévy subordinator.{{cite web|title=Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes|url=http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf|author=Applebaum, D.|pages=37–53|publisher=University of Sheffield}} The Lévy subordinator is a process associated with a Lévy distribution having location parameter of $0$ and a scale parameter of $t^2/2$ . 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.

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:

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

The marginal probability distribution of the symmetric Cauchy process is the Cauchy distribution whose density is{{cite book|title=Probability and Stochastics|url=https://archive.org/details/probabilitystoch00inla_992|url-access=limited|author=Cinlar, E.|page=[https://archive.org/details/probabilitystoch00inla_992/page/n346 332]|year=2011|publisher=Springer|isbn=9780387878591}}{{cite book|title=Essentials of Stochastic Processes|author=Itô, K.|page=54|publisher=American Mathematical Society|year=2006|isbn=9780821838983}}

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

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. In the case where $|\beta|=1$ the process is considered a completely asymmetric Cauchy process.

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 $B>0$ .

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

The characteristic function of the asymmetric Cauchy distribution has the form:

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

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

Category:Lévy processes