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:
- It is a Lévy process[3][4][5]
- It is a stable process[1][2]
- It is a pure jump process[6]
- 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 [math]\displaystyle{ 0 }[/math] and a scale parameter of [math]\displaystyle{ t^2/2 }[/math].[7] The Lévy distribution is a special case of the inverse-gamma distribution. So, using [math]\displaystyle{ C }[/math] to represent the Cauchy process and [math]\displaystyle{ L }[/math] to represent the Lévy subordinator, the symmetric Cauchy process can be described as:
- [math]\displaystyle{ C(t; 0, 1) \;:=\;W(L(t; 0, t^2/2)). }[/math]
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 [math]\displaystyle{ (0,0, W) }[/math], where [math]\displaystyle{ W(dx) = dx / (\pi x^2) }[/math].[8]
The marginal characteristic function of the symmetric Cauchy process has the form:[1][8]
- [math]\displaystyle{ \operatorname{E}\Big[e^{i\theta X_t} \Big] = e^{-t |\theta |}. }[/math]
The marginal probability distribution of the symmetric Cauchy process is the Cauchy distribution whose density is[8][9]
- [math]\displaystyle{ f(x; t) = { 1 \over \pi } \left[ { t \over x^2 + t^2 } \right]. }[/math]
Asymmetric Cauchy process
The asymmetric Cauchy process is defined in terms of a parameter [math]\displaystyle{ \beta }[/math]. Here [math]\displaystyle{ \beta }[/math] is the skewness parameter, and its absolute value must be less than or equal to 1.[1] In the case where [math]\displaystyle{ |\beta|=1 }[/math] the process is considered a completely asymmetric Cauchy process.[1]
The Lévy–Khintchine triplet has the form [math]\displaystyle{ (0,0, W) }[/math], where [math]\displaystyle{ W(dx) = \begin{cases} Ax^{-2}\,dx & \text{if } x\gt 0 \\ Bx^{-2}\,dx & \text{if } x\lt 0 \end{cases} }[/math], where [math]\displaystyle{ A \ne B }[/math], [math]\displaystyle{ A\gt 0 }[/math] and [math]\displaystyle{ B\gt 0 }[/math].[1]
Given this, [math]\displaystyle{ \beta }[/math] is a function of [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math].
The characteristic function of the asymmetric Cauchy distribution has the form:[1]
- [math]\displaystyle{ \operatorname{E}\Big[e^{i\theta X_t} \Big] = e^{-t (|\theta | + i \beta \theta \ln|\theta| / (2 \pi))}. }[/math]
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
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Kovalenko, I.N. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701.
- ↑ 2.0 2.1 Engelbert, H.J., Kurenok, V.P. & Zalinescu, A. (2006). "On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes". From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift. Springer. p. 228. ISBN 9783540307884. https://archive.org/details/fromstochasticca00kaba_517.
- ↑ Winkel, M.. "Introduction to Levy processes". pp. 15–16. http://www.stats.ox.ac.uk/~winkel/lp1.pdf.
- ↑ Jacob, N. (2005). Pseudo Differential Operators & Markov Processes: Markov Processes And Applications, Volume 3. Imperial College Press. p. 135. ISBN 9781860945687.
- ↑ Bertoin, J. (2001). "Some elements on Lévy processes". in Shanbhag, D.N.. Stochastic Processes: Theory and Methods. Gulf Professional Publishing. p. 122. ISBN 9780444500144.
- ↑ Kroese, D.P.; Taimre, T.; Botev, Z.I. (2011). Handbook of Monte Carlo Methods. John Wiley & Sons. p. 214. ISBN 9781118014950. https://archive.org/details/handbookmontecar00kroe.
- ↑ 7.0 7.1 7.2 Applebaum, D.. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes". University of Sheffield. pp. 37–53. http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf.
- ↑ 8.0 8.1 8.2 Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591. https://archive.org/details/probabilitystoch00inla_992.
- ↑ Itô, K. (2006). Essentials of Stochastic Processes. American Mathematical Society. p. 54. ISBN 9780821838983.
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