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:

1. It is a Lévy process^[3][4][5]
2. It is a stable process^[1][2]
3. It is a pure jump process^[6]
4. 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 ${\displaystyle 0}$ and a scale parameter of ${\displaystyle t^{2}/2}$.^[7] The Lévy distribution is a special case of the inverse-gamma distribution. So, using ${\displaystyle C}$ to represent the Cauchy process and ${\displaystyle L}$ to represent the Lévy subordinator, the symmetric Cauchy process can be described as:

${\displaystyle C(t;0,1)\;:=\;W(L(t;0,t^{2}/2)).}$

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.^[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 ${\displaystyle (0,0,W)}$, where ${\displaystyle W(dx)=dx/(\pi x^{2})}$.^[8]

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

${\displaystyle \operatorname {E} {\Big [}e^{i\theta X_{t}}{\Big ]}=e^{-t|\theta |}.}$

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

${\displaystyle f(x;t)={1 \over \pi }\left[{t \over x^{2}+t^{2}}\right].}$

Asymmetric Cauchy process

The asymmetric Cauchy process is defined in terms of a parameter ${\displaystyle \beta }$. Here ${\displaystyle \beta }$ is the skewness parameter, and its absolute value must be less than or equal to 1.^[1] In the case where ${\displaystyle |\beta |=1}$ the process is considered a completely asymmetric Cauchy process.^[1]

The Lévy–Khintchine triplet has the form ${\displaystyle (0,0,W)}$, where ${\displaystyle W(dx)={\begin{cases}Ax^{-2}\,dx&{\text{if }}x>0\\Bx^{-2}\,dx&{\text{if }}x<0\end{cases}}}$, where ${\displaystyle A\neq B}$, ${\displaystyle A>0}$ and ${\displaystyle B>0}$.^[1]

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

The characteristic function of the asymmetric Cauchy distribution has the form:^[1]

${\displaystyle \operatorname {E} {\Big [}e^{i\theta X_{t}}{\Big ]}=e^{-t(|\theta |+i\beta \theta \ln |\theta |/(2\pi ))}.}$

The marginal probability distribution of the asymmetric Cauchy process is a stable distribution with index of stability (i.e., α parameter) equal to 1.