“Cauchy process”的意思、由来-开放百科全书

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]

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

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

.^[12]

The marginal characteristic function of the symmetric Cauchy process has the form:^[1]^[12]

The marginal probability distribution of the symmetric Cauchy process is the Cauchy distribution whose density is^[8]^[9]

Asymmetric Cauchy process

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 equal to 1.

References

1. ^¹²³⁴⁵⁶{{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}}
2. ^¹{{cite book|title=From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift|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=228|year=2006|publisher=Springer|isbn=9783540307884}}
3. ^{{cite web|title=Introduction to Levy processes|author=Winkel, M.|pages=15–16|url=http://www.stats.ox.ac.uk/~winkel/lp1.pdf|accessdate=2013-02-07}}
4. ^{{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}}
5. ^{{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}}
6. ^{{cite book|title=Handbook of Monte Carlo Methods|author1=Kroese, D.P. |author2=Taimre, T. |author3=Botev, Z.I. |page=214|year=2011|publisher=John Wiley & Sons|isbn=9781118014950}}
7. ^¹²{{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}}
8. ^¹²{{cite book|title=Probability and Stochastics|author=Cinlar, E.|page=332|year=2011|publisher=Springer|isbn=9780387878591}}
9. ^{{cite book|title=Essentials of Stochastic Processes|author=Itô, K.|page=54|publisher=American Mathematical Society|year=2006|isbn=9780821838983}}