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:

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 $W(dx)=dx/(\pi x^{2})$ .^[8]

The marginal characteristic function of the symmetric Cauchy process has the form:^[1]^[8]

\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^[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 $W(dx)={\begin{cases}Ax^{-2}\,dx&{\text{if }}x>0\\Bx^{-2}\,dx&{\text{if }}x<0\end{cases}}$ , where $A\neq B$ , $A>0$ 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} {\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

^ ^a ^b ^c ^d ^e ^f ^g Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701.
^ ^a ^b Engelbert, H.J., Kurenok, V.P. & Zalinescu, A. (2006). "On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes". In Kabanov, Y.; Liptser, R.; Stoyanov, J. (eds.). From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift. Springer. p. 228. ISBN 9783540307884.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Winkel, M. "Introduction to Levy processes" (PDF). pp. 15–16. Retrieved 2013-02-07.
^ 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. (ed.). 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.
^ ^a ^b ^c Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.
^ ^a ^b ^c Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591.
^ Itô, K. (2006). Essentials of Stochastic Processes. American Mathematical Society. p. 54. ISBN 9780821838983.

[models-1] ^ ^a ^b ^c ^d ^e ^f ^g Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701.

[stable-2] Engelbert, H.J., Kurenok, V.P. & Zalinescu, A. (2006). "On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes". In Kabanov, Y.; Liptser, R.; Stoyanov, J. (eds.). From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift. Springer. p. 228. ISBN 9783540307884.{{cite book}}: CS1 maint: multiple names: authors list (link)

[Winkel-3] Winkel, M. "Introduction to Levy processes" (PDF). pp. 15–16. Retrieved 2013-02-07.

[4] Jacob, N. (2005). Pseudo Differential Operators & Markov Processes: Markov Processes And Applications, Volume 3. Imperial College Press. p. 135. ISBN 9781860945687.

[5] Bertoin, J. (2001). "Some elements on Lévy processes". In Shanbhag, D.N. (ed.). Stochastic Processes: Theory and Methods. Gulf Professional Publishing. p. 122. ISBN 9780444500144.

[6] Kroese, D.P.; Taimre, T.; Botev, Z.I. (2011). Handbook of Monte Carlo Methods. John Wiley & Sons. p. 214. ISBN 9781118014950.

[applebaum-7] Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.

[cinlar-8] Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591.

[essentials-9] Itô, K. (2006). Essentials of Stochastic Processes. American Mathematical Society. p. 54. ISBN 9780821838983.

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