Cauchy process

The Cauchy process has a number of properties:

1. It is a Lévy process
2. It is a stable process
3. It is a pure jump process
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. The Lévy subordinator is a process associated with a Lévy distribution having location parameter of ${\displaystyle 0}$ and a scale parameter of ${\displaystyle t^2/2}$. 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.

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 ${\displaystyle W(dx)=dx/(\pi x^2)}$.

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

${\displaystyle \mathrm{E}[e^{i\theta X_t}]=e^{-t|\theta |}.}$

The marginal probability distribution of the symmetric Cauchy process is the Cauchy distribution whose density is

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