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Symmetric Cauchy process

The symmetric Cauchy process can be described by a Brownian motion or Wiener process subject to a Lévy subordinator.⁹ The Lévy subordinator is a process associated with a Lévy distribution having location parameter of 0 {\displaystyle 0} and a scale parameter of t 2 / 2 {\displaystyle t^{2}/2} .¹⁰ The Lévy distribution is a special case of the inverse-gamma distribution. So, using C {\displaystyle C} to represent the Cauchy process and L {\displaystyle 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 ) ) . {\displaystyle 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.¹¹

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 ) {\displaystyle (0,0,W)} , where W ( d x ) = d x / ( π x 2 ) {\displaystyle W(dx)=dx/(\pi x^{2})} .¹²

The marginal characteristic function of the symmetric Cauchy process has the form:¹³¹⁴

E ⁡ [ e i θ X t ] = e − t | θ | . {\displaystyle \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 is¹⁵¹⁶

f ( x ; t ) = 1 π [ t x 2 + t 2 ] . {\displaystyle 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 β {\displaystyle \beta } . Here β {\displaystyle \beta } is the skewness parameter, and its absolute value must be less than or equal to 1.¹⁷ In the case where | β | = 1 {\displaystyle |\beta |=1} the process is considered a completely asymmetric Cauchy process.¹⁸

The Lévy–Khintchine triplet has the form ( 0 , 0 , W ) {\displaystyle (0,0,W)} , where W ( d x ) = { A x − 2 d x if  x > 0 B x − 2 d x if  x < 0 {\displaystyle W(dx)={\begin{cases}Ax^{-2}\,dx&{\text{if }}x>0\\Bx^{-2}\,dx&{\text{if }}x<0\end{cases}}} , where A ≠ B {\displaystyle A\neq B} , A > 0 {\displaystyle A>0} and B > 0 {\displaystyle B>0} .¹⁹

Given this, β {\displaystyle \beta } is a function of A {\displaystyle A} and B {\displaystyle B} .

The characteristic function of the asymmetric Cauchy distribution has the form:²⁰

E ⁡ [ e i θ X t ] = e − t ( | θ | + i β θ ln ⁡ | θ | / ( 2 π ) ) . {\displaystyle \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

1. Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. 9780849328701 ↩

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) 9783540307884 ↩

3. Winkel, M. "Introduction to Levy processes" (PDF). pp. 15–16. Retrieved 2013-02-07. http://www.stats.ox.ac.uk/~winkel/lp1.pdf ↩

4. Jacob, N. (2005). Pseudo Differential Operators & Markov Processes: Markov Processes And Applications, Volume 3. Imperial College Press. p. 135. ISBN 9781860945687. 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. 9780444500144 ↩

6. Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. 9780849328701 ↩

7. 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) 9783540307884 ↩

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

9. Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53. http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf ↩

10. Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53. http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf ↩

11. Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53. http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf ↩

12. Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591. 9780387878591 ↩

13. Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. 9780849328701 ↩

14. Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591. 9780387878591 ↩

15. Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591. 9780387878591 ↩

16. Itô, K. (2006). Essentials of Stochastic Processes. American Mathematical Society. p. 54. ISBN 9780821838983. 9780821838983 ↩

17. Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. 9780849328701 ↩

18. Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. 9780849328701 ↩

19. Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. 9780849328701 ↩

20. Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. 9780849328701 ↩