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 symmetric Cauchy process can be described by a Brownian motion or Wiener process subject to a Lévysubordinator.[7] The Lévy subordinator is a process associated with a Lévy distribution having location parameter of and a scale parameter of .[7] The Lévy distribution is a special case of the inverse-gamma distribution. So, using to represent the Cauchy process and to represent the Lévy subordinator, the symmetric Cauchy process can be described as:
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 , where .[8]
The asymmetric Cauchy process is defined in terms of a parameter . Here
is the skewness parameter, and its absolute value must be less than or equal to 1.[1] In the case where the process is considered a completely asymmetric Cauchy process.[1]
The Lévy–Khintchine triplet has the form , where , where , and .[1]
Given this, is a function of and .
The characteristic function of the asymmetric Cauchy distribution has the form:[1]
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
^ abcdefgKovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN9780849328701.
^Jacob, N. (2005). Pseudo Differential Operators & Markov Processes: Markov Processes And Applications, Volume 3. Imperial College Press. p. 135. ISBN9781860945687.
^Bertoin, J. (2001). "Some elements on Lévy processes". In Shanbhag, D.N. (ed.). Stochastic Processes: Theory and Methods. Gulf Professional Publishing. p. 122. ISBN9780444500144.