stable count distribution

Its standard distribution is defined as

: $\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)=\backslash frac\{1\}\{\backslash Gamma(\backslash frac\{1\}\{\backslash alpha\}+1)\}\; \backslash frac\{1\}\{\backslash nu\}\; L\_\backslash alpha\backslash left(\backslash frac\{1\}\{\backslash nu\}\backslash right),$

where $\backslash nu>0$ and

Its location-scale family is defined as

: $\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu;\backslash nu\_0,\backslash theta)=\; \backslash frac\{1\}\{\backslash Gamma(\backslash frac\{1\}\{\backslash alpha\}+1)\}$

\frac{1}{\nu-\nu_0} L_\alpha\left(\frac{\theta}{\nu-\nu_0}\right),

where $\backslash nu>\backslash nu\_0$, $\backslash theta>0$, and

In the above expression, $L\_\backslash alpha(x)$ is a one-sided stable distribution,{{Cite journal|last1=Penson|first1=K. A.|last2=Górska|first2=K.|date=2010-11-17|title=Exact and Explicit Probability Densities for One-Sided Lévy Stable Distributions|journal=Physical Review Letters|volume=105|issue=21|pages=210604|arxiv=1007.0193|bibcode=2010PhRvL.105u0604P|doi=10.1103/PhysRevLett.105.210604|pmid=21231282|s2cid=27497684}} which is defined as following.

Let $X$ be a standard stable random variable whose distribution is characterized by $f(x;\backslash alpha,\backslash beta,c,\backslash mu)$, then we have

: $L\_\backslash alpha(x)=f(x;\backslash alpha,1,\backslash cos\backslash left(\backslash frac\{\backslash pi\backslash alpha\}\{2\}\backslash right)^\{1/\backslash alpha\},0),$

where .

Consider the Lévy sum $Y\; =\; \backslash sum\_\{i=1\}^N\; X\_i$ where $X\_i\backslash sim\; L\_\backslash alpha(x)$, then $Y$

has the density $\backslash frac\{1\}\{\backslash nu\}\; L\_\backslash alpha\backslash left(\backslash frac\{x\}\{\backslash nu\}\backslash right)$ where $\backslash nu=N^\{1/\backslash alpha\}$. Set $x=1$, we arrive at $\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)$

without the normalization constant.

The reason why this distribution is called "stable count" can be understood by the relation $\backslash nu=N^\{1/\backslash alpha\}$. Note that $N$ is the "count" of the Lévy sum. Given a fixed $\backslash alpha$, this distribution gives the probability of taking $N$ steps to travel one unit of distance.

Based on the integral form of $L\_\backslash alpha(x)$ and $q=\backslash exp(-i\backslash alpha\backslash pi/2)$

, we have the integral form of $\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)$ as

:$\backslash begin\{align\}\; \backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)$

& = \frac{2}{\pi\Gamma(\frac{1}{\alpha}+1)} \int_0^\infty e^{-\text{Re}(q)\,t^\alpha}

\frac{1}{\nu} \sin(\frac{t}{\nu})\sin(-\text{Im}(q)\,t^\alpha) \,dt, \text{ or }

\\ & = \frac{2}{\pi\Gamma(\frac{1}{\alpha}+1)} \int_0^\infty e^{-\text{Re}(q)\,t^\alpha}

\frac{1}{\nu} \cos(\frac{t}{\nu})\cos(\text{Im}(q)\,t^\alpha) \,dt .

\\ \end{align}

Based on the double-sine integral above, it leads to the integral form of the standard CDF:

:$\backslash begin\{align\}\; \backslash Phi\_\backslash alpha(x)$

& = \frac{2}{\pi\Gamma(\frac{1}{\alpha}+1)} \int_0^x \int_0^\infty e^{-\text{Re}(q)\,t^\alpha}

\frac{1}{\nu} \sin(\frac{t}{\nu})\sin(-\text{Im}(q)\,t^\alpha) \,dt\,d\nu

\\ & = 1- \frac{2}{\pi\Gamma(\frac{1}{\alpha}+1)} \int_0^\infty e^{-\text{Re}(q)\,t^\alpha}

\sin(-\text{Im}(q)\,t^\alpha) \,\text{Si}(\frac{t}{x}) \,dt,

\\ \end{align}

where $\backslash text\{Si\}(x)=\backslash int\_0^x\; \backslash frac\{\backslash sin(x)\}\{x\}\backslash ,dx$ is the sine integral function.

In "Series representation", it is shown that the stable count distribution is a special case of the Wright function (See Section 4 of {{Cite SSRN|last=Lihn|first=Stephen|date=2020|title=Stable Count Distribution for the Volatility Indices and Space-Time Generalized Stable Characteristic Function|ssrn=3659383}}):

:$\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)\; =$

\frac{1}{\Gamma\left( \frac{1}{\alpha}+1 \right)}

W_{-\alpha,0}(-\nu^\alpha)

, \, \text{where} \,\,

W_{\lambda,\mu}(z) = \sum_{n=0}^\infty

\frac{z^n}{n!\,\Gamma(\lambda n+\mu)}.

This leads to the Hankel integral: (based on (1.4.3) of {{Cite book|title=Fractional and Multivariable Calculus|last1=Mathai|first1=A.M.|last2=Haubold|first2=H.J.|date=2017|publisher=Springer International Publishing|isbn=9783319599922|series=Springer Optimization and Its Applications|volume=122|location=Cham|doi=10.1007/978-3-319-59993-9}})

:$\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)\; =$

\frac{1}{\Gamma\left( \frac{1}{\alpha}+1 \right)}

\frac{1}{2 \pi i}

\int_{Ha} e^{t-(\nu t)^\alpha} \, dt,

\,

where Ha represents a Hankel contour.

Another approach to derive the stable count distribution is to use the Laplace transform of the one-sided stable distribution, (Section 2.4 of )

: $\backslash int\_0^\backslash infty\; e^\{-z\; x\}\; L\_\backslash alpha(x)\; \backslash ,\; dx\; =\; e^\{-z^\backslash alpha\},$where .

Let $x=1/\backslash nu$, and one can decompose the integral on the left hand side as a product distribution of a standard Laplace distribution and a standard stable count distribution,

:$\backslash frac\{1\}\{2\}\; \backslash frac\{1\}\{\backslash Gamma(\backslash frac\{1\}\{\backslash alpha\}+1)\}\; e^\{-|z|^\backslash alpha\}\; =$

\int_0^\infty \frac{1}{\nu} \left( \frac{1}{2} e^{-|z|/\nu} \right)

\left(\frac{1}{\Gamma(\frac{1}{\alpha}+1)} \frac{1}{\nu} L_\alpha \left( \frac{1}{\nu} \right) \right) \, d\nu

=

\int_0^\infty \frac{1}{\nu} \left( \frac{1}{2} e^{-|z|/\nu} \right)

\mathfrak{N}_\alpha(\nu) \, d\nu

,

where $z\; \backslash in\; \backslash mathsf\{R\}$.

This is called the "lambda decomposition" (See Section 4 of ) since the LHS was named as "symmetric lambda distribution" in Lihn's former works. However, it has several more popular names such as "exponential power distribution", or the "generalized error/normal distribution", often referred to when $\backslash alpha>1$. It is also the Weibull survival function in Reliability engineering.

Lambda decomposition is the foundation of Lihn's framework of asset returns under the stable law. The LHS is the distribution of asset returns. On the RHS, the Laplace distribution represents the lepkurtotic noise, and the stable count distribution represents the volatility.

A variant of the stable count distribution is called the stable vol distribution $V\_\{\backslash alpha\}(s)$.

The Laplace transform of $e^\{-|z|^\backslash alpha\}$ can be re-expressed in terms of a Gaussian mixture of $V\_\{\backslash alpha\}(s)$ (See Section 6 of ).

It is derived from the lambda decomposition above by a change of variable such that

:$\backslash frac\{1\}\{2\}\; \backslash frac\{1\}\{\backslash Gamma(\backslash frac\{1\}\{\backslash alpha\}+1)\}\; e^\{-|z|^\backslash alpha\}\; =$

\frac{1}{2} \frac{1}{\Gamma(\frac{1}{\alpha}+1)} e^{-(z^2)^{\alpha/2}} =

\int_0^\infty \frac{1}{s} \left( \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} (z/s)^2} \right) V_{\alpha}(s) \, ds

,

where

:$$

\begin{align}

V_{\alpha}(s) &= \displaystyle

\frac{\sqrt{2 \pi} \,\Gamma(\frac{2}{\alpha}+1)}{\Gamma(\frac{1}{\alpha}+1)} \,

\mathfrak{N}_{\frac{\alpha}{2}}(2 s^2), \,\, 0 < \alpha \leq 2

\\

&= \displaystyle

\frac{ \sqrt{2\pi} }{ \Gamma(\frac{1}{\alpha}+1) }

\, W_{-\frac{\alpha}{2},0} \left( -{(\sqrt{2} s)}^\alpha \right)

\end{align}

This transformation is named generalized Gauss transmutation since it generalizes the [https://arxiv.org/abs/1510.08765 Gauss-Laplace transmutation], which is equivalent to $V\_\{1\}(s)\; =\; 2\; \backslash sqrt\{2\; \backslash pi\}\; \backslash ,\; \backslash mathfrak\{N\}\_\{\backslash frac\{1\}\{2\}\}(2\; s^2)\; =\; s\; \backslash ,\; e^\{-s^2/2\}$.

The shape parameter of the Gamma and Poisson Distributions is connected to the inverse of Lévy's stability parameter $1/\backslash alpha$.

The upper regularized gamma function $Q(s,x)$ can be expressed as an incomplete integral of $e^\{-\{u^\backslash alpha\}\}$ as

Q(\frac{1}{\alpha}, z^\alpha) =

\frac{1}{\Gamma(\frac{1}{\alpha}+1)}

\displaystyle\int_z^\infty

e^{-{u^\alpha}} \, du.

By replacing $e^\{-\{u^\backslash alpha\}\}$ with the decomposition and carrying out one integral, we have:

Q(\frac{1}{\alpha}, z^\alpha) =

\displaystyle\int_z^\infty \, du

\displaystyle\int_0^\infty

\frac{1}{\nu} \left( e^{-u/\nu} \right)

\, \mathfrak{N}_{\alpha}\left(\nu\right) \, d\nu

= \displaystyle\int_0^\infty

\left( e^{-z/\nu} \right)

\, \mathfrak{N}_{\alpha}\left(\nu\right) \, d\nu.

Reverting $(\backslash frac\{1\}\{\backslash alpha\},\; z^\backslash alpha)$ back to $(s,x)$, we arrive at the decomposition of $Q(s,x)$ in terms of a stable count:

Q(s,x) =

\displaystyle\int_0^\infty e^{\left( -{x^s}/{\nu} \right)}

\, \mathfrak{N}_{{1}/{s}}\left(\nu\right) \, d\nu. \,\, (s > 1)

Differentiate $Q(s,x)$ by $x$, we arrive at the desired formula:

:$$

\begin{align}

\frac{1}{\Gamma(s)} x^{s-1} e^{-x} & =

\displaystyle\int_0^\infty

\frac{1}{\nu} \left[ s\, x^{s-1} e^{\left( -{x^s}/{\nu} \right)} \right]

\, \mathfrak{N}_{{1}/{s}}\left(\nu\right) \, d\nu

\\

& =

\displaystyle\int_0^\infty

\frac{1}{t} \left[ s\, {\left( \frac{x}{t} \right)}^{s-1} e^{-{\left( x/t \right)}^s} \right]

\, \left[ \mathfrak{N}_{{1}/{s}}\left(t^s\right) \, s \, t^{s-1} \right] \, dt

\,\,\, (\nu = t^s)

\\

& =

\displaystyle\int_0^\infty

\frac{1}{t} \, \text{Weibull}\left( \frac{x}{t}; s\right)

\, \left[ \mathfrak{N}_{{1}/{s}}\left(t^s\right) \, s \, t^{s-1} \right] \, dt

\end{align}

This is in the form of a product distribution. The term $\backslash left[\; s\backslash ,\; \{\backslash left(\; \backslash frac\{x\}\{t\}\; \backslash right)\}^\{s-1\}\; e^\{-\{\backslash left(\; x/t\; \backslash right)\}^s\}\; \backslash right]$ in the RHS is associated with a Weibull distribution of shape $s$. Hence, this formula connects the stable count distribution to the probability density function of a Gamma distribution (here) and the probability mass function of a Poisson distribution (here, $s\; \backslash rightarrow\; s+1$). And the shape parameter $s$ can be regarded as inverse of Lévy's stability parameter $1/\backslash alpha$.

The degrees of freedom $k$ in the chi and chi-squared Distributions can be shown to be related to $2/\backslash alpha$. Hence, the original idea of viewing $\backslash lambda\; =\; 2/\backslash alpha$ as an integer index in the lambda decomposition is justified here.

For the chi-squared distribution, it is straightforward since the chi-squared distribution is a special case of the gamma distribution, in that $\backslash chi^2\_k\; \backslash sim\; \backslash text\{Gamma\}\; \backslash left(\backslash frac\{k\}\{2\},\; \backslash theta=2\; \backslash right)$. And from above, the shape parameter of a gamma distribution is $1/\backslash alpha$.

For the chi distribution, we begin with its CDF $P\; \backslash left(\; \backslash frac\{k\}2,\; \backslash frac\{x^2\}2\; \backslash right)$, where $P(s,x)\; =\; 1\; -\; Q(s,x)$. Differentiate $P\; \backslash left(\; \backslash frac\{k\}2,\; \backslash frac\{x^2\}2\; \backslash right)$ by $x$ , we have its density function as

: $$

\begin{align}

\chi_k(x) =

\frac{x^{k-1} e^{-x^2/2}}

{2^{\frac{k}2-1} \Gamma \left( \frac{k}2 \right)}

& =

\displaystyle\int_0^\infty

\frac{1}{\nu} \left[ 2^{-\frac{k}2} \,k \, x^{k-1} e^{\left( -2^{-\frac{k}2} \, {x^k}/{ \nu} \right)} \right]

\, \mathfrak{N}_{\frac{2}{k}}\left(\nu\right) \, d\nu

\\

& =

\displaystyle\int_0^\infty

\frac{1}{t} \left[ k\, {\left( \frac{x}{t} \right)}^{k-1} e^{-{\left( x/t \right)}^k} \right]

\, \left[ \mathfrak{N}_{\frac{2}{k}}\left( 2^{-\frac{k}2} t^k \right) \, 2^{-\frac{k}2} \, k \, t^{k-1} \right] \, dt,

\,\,\, (\nu = 2^{-\frac{k}2} t^k)

\\

& =

\displaystyle\int_0^\infty

\frac{1}{t} \, \text{Weibull}\left( \frac{x}{t}; k\right)

\, \left[ \mathfrak{N}_{\frac{2}{k}}\left( 2^{-\frac{k}2} t^k \right) \, 2^{-\frac{k}2} \, k \, t^{k-1} \right] \, dt

\end{align}

This formula connects $2/k$ with $\backslash alpha$ through the $$

\mathfrak{N}_{\frac{2}{k}}\left( \cdot \right)

term.

The generalized gamma distribution is a probability distribution with two shape parameters, and is the super set of the gamma distribution, the Weibull distribution, the exponential distribution, and the half-normal distribution. Its CDF is in the form of $P(s,\; x^c)\; =\; 1\; -\; Q(s,\; x^c)$.

(Note: We use $s$ instead of $a$ for consistency and to avoid confusion with $\backslash alpha$.)

Differentiate $P(s,x^c)$ by $x$, we arrive at the product-distribution formula:

:$$

\begin{align}

\text{GenGamma}(x; s, c) & =

\displaystyle\int_0^\infty

\frac{1}{t} \, \text{Weibull}\left( \frac{x}{t}; sc\right)

\, \left[ \mathfrak{N}_{\frac{1}{s}}\left(t^{sc}\right) \, sc \, t^{sc-1} \right] \, dt

\,\, (s \geq 1)

\end{align}

where $\backslash text\{GenGamma\}(x;\; s,\; c)$ denotes the PDF of a generalized gamma distribution,

whose CDF is parametrized as $P(s,x^c)$.

This formula connects $1/s$ with $\backslash alpha$ through the $$

\mathfrak{N}_{\frac{1}{s}}\left( \cdot \right)

term. The $sc$ term is an exponent representing the second degree of freedom in the shape-parameter space.

This formula is singular for the case of a Weibull distribution since $s$ must be one for $\backslash text\{GenGamma\}(x;\; 1,\; c)\; =\; \backslash text\{Weibull\}(x;\; c)$;

but for $\backslash mathfrak\{N\}\_\{\backslash frac\{1\}\{s\}\}\backslash left(\backslash nu\backslash right)$ to exist, $s$ must be greater than one.

When $s\backslash rightarrow\; 1$, $\backslash mathfrak\{N\}\_\{\backslash frac\{1\}\{s\}\}\backslash left(\backslash nu\backslash right)$ is a delta function and this formula becomes trivial.

The Weibull distribution has its distinct way of decomposition as following.

For a Weibull distribution whose CDF is $F(x;k,\backslash lambda)\; =\; 1\; -\; e^\{-(x/\backslash lambda)^k\}\; \backslash ,\backslash ,\; (x>0)$, its shape parameter $k$ is equivalent to Lévy's stability parameter $\backslash alpha$.

A similar expression of product distribution can be derived, such that the kernel is either

a one-sided Laplace distribution $F(x;1,\backslash sigma)$ or a Rayleigh distribution $F(x;2,\backslash sqrt\{2\}\; \backslash sigma)$.

It begins with the complementary CDF, which comes from Lambda decomposition:

:$$

1-F(x;k,1) =

\begin{cases}

\displaystyle\int_0^\infty \frac{1}{\nu} \, (1-F(x;1,\nu))

\left[ \Gamma \left( \frac{1}{k}+1 \right) \mathfrak{N}_k(\nu) \right] \, d\nu ,

& 1 \geq k > 0; \text{or } \\

\displaystyle\int_0^\infty \frac{1}{s} \, (1-F(x;2,\sqrt{2} s))

\left[ \sqrt{\frac{2}{\pi}} \, \Gamma \left( \frac{1}{k}+1 \right) V_k(s) \right] \, ds ,

& 2 \geq k > 0.

\end{cases}

By taking derivative on $x$, we obtain the product distribution form of a Weibull distribution PDF $\backslash text\{Weibull\}(x;k)$ as

:$$

\text{Weibull}(x;k) =

\begin{cases}

\displaystyle\int_0^\infty \frac{1}{\nu} \, \text{Laplace}(\frac{x}{\nu})

\left[ \Gamma \left( \frac{1}{k}+1 \right) \frac{1}{\nu} \mathfrak{N}_k(\nu) \right] \, d\nu ,

& 1 \geq k > 0; \text{or } \\

\displaystyle\int_0^\infty \frac{1}{s} \, \text{Rayleigh}(\frac{x}{s})

\left[ \sqrt{\frac{2}{\pi}} \, \Gamma \left( \frac{1}{k}+1 \right) \frac{1}{s} V_k(s) \right] \, ds ,

& 2 \geq k > 0.

\end{cases}

where $\backslash text\{Laplace\}(x)\; =\; e^\{-x\}$ and $\backslash text\{Rayleigh\}(x)\; =\; x\; e^\{-x^2/2\}$.

it is clear that $k\; =\; \backslash alpha$ from the $\backslash mathfrak\{N\}\_k(\backslash nu)$ and $V\_k(s)$ terms.

For stable distribution family, it is essential to understand its asymptotic behaviors. From, for small $\backslash nu$,

:$\backslash begin\{align\}\; \backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)$

& \rightarrow B(\alpha) \,\nu^{\alpha}, \text{ for } \nu \rightarrow 0 \text{ and } B(\alpha)>0.

\\ \end{align}

This confirms $\backslash mathfrak\{N\}\_\backslash alpha(0)=0$

.

For large $\backslash nu$,

:$\backslash begin\{align\}\; \backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)$

& \rightarrow \nu^{\frac{\alpha}{2(1-\alpha)}} e^{-A(\alpha) \,\nu^{\frac{\alpha}{1-\alpha}}},

\text{ for } \nu \rightarrow \infty \text{ and } A(\alpha)>0.

\\ \end{align}

This shows that the tail of $\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)$

decays exponentially at infinity. The larger $\backslash alpha$

is, the stronger the decay.

This tail is in the form of a generalized gamma distribution, where in its $f(x;\; a,\; d,\; p)$ parametrization,

$p\; =\; \backslash frac\{\backslash alpha\}\{1-\backslash alpha\}$, $a\; =\; A(\backslash alpha)^\{-1/p\}$, and $d\; =\; 1\; +\; \backslash frac\{p\}\{2\}$.

Hence, it is equivalent to $\backslash text\{GenGamma\}(\backslash frac\{x\}\{a\};\; s\; =\; \backslash frac\{1\}\{\backslash alpha\}\; -\backslash frac\{1\}\{2\},\; c\; =\; p)$,

whose CDF is parametrized as $P\backslash left(\; s,\backslash left(\; \backslash frac\{x\}\{a\}\; \backslash right)^c\; \backslash right)$.

The n-th moment $m\_n$ of $\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)$

is the $-(n+1)$-th moment of $L\_\backslash alpha(x)$. All positive moments are finite. This in a way solves the thorny issue of diverging moments in the stable distribution. (See Section 2.4 of )

:$$

\begin{align} m_n &

= \int_0^\infty \nu^n \mathfrak{N}_\alpha(\nu) d\nu

= \frac{1}{\Gamma(\frac{1}{\alpha}+1)} \int_0^\infty \frac{1}{t^{n+1}} L_\alpha(t) \, dt.

\\ \end{align}

The analytic solution of moments is obtained through the Wright function:

:$$

\begin{align} m_n &

= \frac{1}{\Gamma(\frac{1}{\alpha}+1)} \int_0^\infty \nu^{n} W_{-\alpha,0}(-\nu^\alpha) \, d\nu

\\ &

= \frac{\Gamma(\frac{n+1}{\alpha})}{\Gamma(n+1)\Gamma(\frac{1}{\alpha})}, \, n \geq -1.

\\ \end{align}

where $\backslash int\_0^\backslash infty\; r^\backslash delta\; W\_\{-\backslash nu,\backslash mu\}(-r)\backslash ,dr\; =$

\frac{\Gamma(\delta+1)}{\Gamma(\nu\delta+\nu+\mu)}

, \, \delta>-1,0<\nu<1,\mu>0. (See (1.4.28) of )

Thus, the mean of $\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)$

is

:$m\_1=\backslash frac\{\backslash Gamma(\backslash frac\{2\}\{\backslash alpha\})\}\{\backslash Gamma(\backslash frac\{1\}\{\backslash alpha\})\}$

The variance is

:$\backslash sigma^2=$

\frac{\Gamma(\frac{3}{\alpha})}{2\Gamma(\frac{1}{\alpha})}

- \left[ \frac{\Gamma(\frac{2}{\alpha})}{\Gamma(\frac{1}{\alpha})} \right]^2

And the lowest moment is $m\_\{-1\}\; =\; \backslash frac\{1\}\{\backslash Gamma(\backslash frac\{1\}\{\backslash alpha\}\; +\; 1)\}$ by applying

$\backslash Gamma(\backslash frac\{x\}\{y\})\; \backslash to\; y\backslash Gamma(x)$ when $x\; \backslash to\; 0$.

The n-th moment of the stable vol distribution $V\_\backslash alpha(s)$ is

:$$

\begin{align}

m_n(V_\alpha) & = 2^{-\frac{n}{2}} \sqrt{\pi} \,

\frac{\Gamma(\frac{n+1}{\alpha})}{\Gamma(\frac{n+1}{2}) \Gamma(\frac{1}{\alpha})}, \, n \geq -1.

\end{align}

The MGF can be expressed by a Fox-Wright function or Fox H-function:

:$\backslash begin\{align\}\; M\_\backslash alpha(s)$

& = \sum_{n=0}^\infty \frac{m_n\,s^n}{n!}

= \frac{1}{\Gamma(\frac{1}{\alpha})} \sum_{n=0}^\infty

\frac{\Gamma(\frac{n+1}{\alpha})\,s^n}{\Gamma(n+1)^2}

\\ & = \frac{1}{\Gamma(\frac{1}{\alpha})} {}_1\Psi_1\left[(\frac{1}{\alpha},\frac{1}{\alpha});(1,1); s\right]

,\,\,\text{or}

\\ & = \frac{1}{\Gamma(\frac{1}{\alpha})} H^{1,1}_{1,2}\left[-s \bigl|

\begin{matrix} (1-\frac{1}{\alpha}, \frac{1}{\alpha}) \\ (0,1);(0,1) \end{matrix}

\right]

\\ \end{align}

As a verification, at $\backslash alpha=\backslash frac\{1\}\{2\}$, $M\_\{\backslash frac\{1\}\{2\}\}(s)\; =\; (1-4s)^\{-\backslash frac\{3\}\{2\}\}$ (see below) can be Taylor-expanded to $\{\}\_1\backslash Psi\_1\backslash left[(2,2);(1,1);\; s\backslash right]$

=\sum_{n=0}^\infty \frac{\Gamma(2n+2)\,s^n}{\Gamma(n+1)^2}

via $\backslash Gamma(\backslash frac\{1\}\{2\}-n)\; =\; \backslash sqrt\{\backslash pi\}\; \backslash frac\{(-4)^n\; n!\}\{(2n)!\}$.

When $\backslash alpha=\backslash frac\{1\}\{2\}$, $L\_\{1/2\}(x)$ is the Lévy distribution which is an inverse gamma distribution. Thus $\backslash mathfrak\{N\}\_\{1/2\}(\backslash nu;\backslash nu\_0,\backslash theta)$ is a shifted gamma distribution of shape 3/2 and scale $4\backslash theta$,

: $\backslash mathfrak\{N\}\_\{\backslash frac\{1\}\{2\}\}(\backslash nu;\backslash nu\_0,\backslash theta)\; =\; \backslash frac\{1\}\{4\backslash sqrt\{\backslash pi\}\backslash theta^\{3/2\}\}\; (\backslash nu-\backslash nu\_0)^\{1/2\}\; e^\{-(\backslash nu-\backslash nu\_0)/4\backslash theta\},$

where $\backslash nu>\backslash nu\_0$, $\backslash theta>0$.

Its mean is $\backslash nu\_0+6\backslash theta$ and its standard deviation is $\backslash sqrt\{24\}\backslash theta$. This called "quartic stable count distribution". The word "quartic" comes from Lihn's former work on the lambda distribution{{Cite SSRN|last=Lihn|first=Stephen H. T.|date=2017-01-26|title=From Volatility Smile to Risk Neutral Probability and Closed Form Solution of Local Volatility Function|ssrn=2906522}} where $\backslash lambda=2/\backslash alpha=4$. At this setting, many facets of stable count distribution have elegant analytical solutions.

The p-th central moments are $\backslash frac\{2\; \backslash Gamma(p+3/2)\}\{\backslash Gamma(3/2)\}\; 4^p\backslash theta^p$. The CDF is $\backslash frac\{2\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash gamma\backslash left(\backslash frac\{3\}\{2\},\; \backslash frac\{\backslash nu-\backslash nu\_0\}\{4\backslash theta\}\; \backslash right)$ where $\backslash gamma(s,x)$ is the lower incomplete gamma function. And the MGF is $M\_\{\backslash frac\{1\}\{2\}\}(s)\; =\; e^\{s\backslash nu\_0\}(1-4s\backslash theta)^\{-\backslash frac\{3\}\{2\}\}$. (See Section 3 of )

As $\backslash alpha$

becomes larger, the peak of the distribution becomes sharper. A special case of $\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)$

is when $\backslash alpha\backslash rightarrow1$. The distribution behaves like a Dirac delta function,

:$\backslash mathfrak\{N\}\_\{\backslash alpha\backslash to\; 1\}(\backslash nu)\; \backslash to\; \backslash delta(\backslash nu-1),$

where

, and $\backslash int\_\{0\_-\}^\{0\_+\}\; \backslash delta(x)\; dx\; =\; 1$

.

Likewise, the stable vol distribution at $\backslash alpha\; \backslash to\; 2$ also becomes a delta function,

:$V\_\{\backslash alpha\backslash to\; 2\}(s)\; \backslash to\; \backslash delta(s-\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}).$

Based on the series representation of the one-sided stable distribution, we have:

:$\backslash begin\{align\}\; \backslash mathfrak\{N\}\_\backslash alpha(x)$

& =

\frac{1}{\pi\Gamma(\frac{1}{\alpha}+1)}

\sum_{n=1}^\infty\frac{-\sin(n(\alpha+1)\pi)}{n!}{x}^{\alpha n}\Gamma(\alpha n+1)

\\ & =

\frac{1}{\pi\Gamma(\frac{1}{\alpha}+1)}

\sum_{n=1}^\infty\frac{(-1)^{n+1} \sin(n\alpha\pi)}{n!}{x}^{\alpha n}\Gamma(\alpha n+1)

\\ \end{align}

.

This series representation has two interpretations:

• First, a similar form of this series was first given in Pollard (1948),{{Cite journal|last=Pollard|first=Harry|date=1948-12-01|title=The completely monotonic character of the Mittag-Leffler function {{math|E_a(−x)}}|journal=Bulletin of the American Mathematical Society|language=en|volume=54|issue=12|pages=1115–1117|doi=10.1090/S0002-9904-1948-09132-7|issn=0002-9904|doi-access=free}} and in "Relation to Mittag-Leffler function", it is stated that $\backslash mathfrak\{N\}\_\backslash alpha(x)\; =$

\frac{\alpha^2 x^\alpha}{\Gamma \left(\frac{1}{\alpha}\right)}

H_\alpha(x^\alpha), where $H\_\backslash alpha(k)$ is the Laplace transform of the Mittag-Leffler function $E\_\backslash alpha(-x)$ .

• Secondly, this series is a special case of the Wright function $W\_\{\backslash lambda,\backslash mu\}(z)$: (See Section 1.4 of )

:$\backslash begin\{align\}\; \backslash mathfrak\{N\}\_\backslash alpha(x)$

& =

\frac{1}{\pi\Gamma(\frac{1}{\alpha}+1)}

\sum_{n=1}^\infty\frac{(-1)^{n} {x}^{\alpha n}}{n!}\,

\sin((\alpha n+1)\pi)\Gamma(\alpha n+1)

\\ & =

\frac{1}{\Gamma \left(\frac{1}{\alpha}+1\right)}

W_{-\alpha,0}(-x^\alpha), \, \text{where} \,\,

W_{\lambda,\mu}(z) = \sum_{n=0}^\infty

\frac{z^n}{n!\,\Gamma(\lambda n+\mu)},

\lambda>-1.

\\ \end{align}

The proof is obtained by the reflection formula of the Gamma function: $\backslash sin((\backslash alpha\; n+1)\backslash pi)\backslash Gamma(\backslash alpha\; n+1)\; =\; \backslash pi/\backslash Gamma(-\backslash alpha\; n)$, which admits the mapping: $\backslash lambda=-\backslash alpha,\backslash mu=0,z=-x^\backslash alpha$

in $W\_\{\backslash lambda,\backslash mu\}(z)$. The Wright representation leads to analytical solutions for many statistical properties of the stable count distribution and establish another connection to fractional calculus.

Stable count distribution can represent the daily distribution of VIX quite well. It is hypothesized that VIX is distributed like $\backslash mathfrak\{N\}\_\{\backslash frac\{1\}\{2\}\}(\backslash nu;\backslash nu\_0,\backslash theta)$

with $\backslash nu\_0=10.4$

and $\backslash theta=1.6$

(See Section 7 of ). Thus the stable count distribution is the first-order marginal distribution of a volatility process. In this context, $\backslash nu\_0$

is called the "floor volatility". In practice, VIX rarely drops below 10. This phenomenon justifies the concept of "floor volatility". A sample of the fit is shown below:

File:Screen_Shot_2019-08-09_at_4.59.58_PM.png

One form of mean-reverting SDE for $\backslash mathfrak\{N\}\_\{\backslash frac\{1\}\{2\}\}(\backslash nu;\backslash nu\_0,\backslash theta)$ is based on a modified Cox–Ingersoll–Ross (CIR) model. Assume $S\_t$ is the volatility process, we have

:$dS\_t\; =\; \backslash frac\{\backslash sigma^2\}\{8\backslash theta\}\; (6\backslash theta+\backslash nu\_0-S\_t)\; \backslash ,\; dt\; +\; \backslash sigma\; \backslash sqrt\{S\_t-\backslash nu\_0\}\; \backslash ,\; dW,$

where $\backslash sigma$

is the so-called "vol of vol". The "vol of vol" for VIX is called VVIX, which has a typical value of about 85.{{Cite web|url=https://cdn.cboe.com/resources/indices/documents/vvix-termstructure.pdf|title=DOUBLE THE FUN WITH CBOE's VVIX Index|last=|first=|date=|website=www.cboe.com|archive-url=|archive-date=|access-date=2019-08-09}}

This SDE is analytically tractable and satisfies [https://quant.stackexchange.com/questions/44475/feller-condition-cox-ingersoll-ross-source the Feller condition], thus $S\_t$ would never go below $\backslash nu\_0$

. But there is a subtle issue between theory and practice. There has been about 0.6% probability that VIX did go below $\backslash nu\_0$

. This is called "spillover". To address it, one can replace the square root term with $\backslash sqrt\{\backslash max(S\_t-\backslash nu\_0,\backslash delta\backslash nu\_0)\}$, where $\backslash delta\backslash nu\_0\backslash approx\; 0.01\; \backslash ,\; \backslash nu\_0$ provides a small leakage channel for $S\_t$ to drift slightly below $\backslash nu\_0$

.

Extremely low VIX reading indicates a very complacent market. Thus the spillover condition, , carries a certain significance - When it occurs, it usually indicates the calm before the storm in the business cycle.

As the modified CIR model above shows, it takes another input parameter $\backslash sigma$ to simulate sequences of stable count random variables. The mean-reverting stochastic process takes the form of

: $dS\_t\; =\; \backslash sigma^2\; \backslash mu\_\{\backslash alpha\}\backslash left(\; \backslash frac\{S\_t\}\{\backslash theta\}\; \backslash right)\; \backslash ,\; dt\; +\; \backslash sigma\; \backslash sqrt\{S\_t\}\; \backslash ,\; dW,$

which should produce $\backslash \{S\_t\backslash \}$ that distributes like $\backslash mathfrak\{N\}\_\{\backslash alpha\}(\backslash nu;\backslash theta)$ as $t\; \backslash rightarrow\; \backslash infty$.

And $\backslash sigma$ is a user-specified preference for how fast $S\_t$ should change.

By solving the Fokker-Planck equation, the solution for $\backslash mu\_\{\backslash alpha\}(x)$ in terms of $\backslash mathfrak\{N\}\_\{\backslash alpha\}(x)$ is

: $\backslash begin\{array\}\{lcl\}$

\mu_\alpha(x) & = & \displaystyle

\frac{1}{2} \frac{\left( x {d \over dx} +1 \right) \mathfrak{N}_{\alpha}(x)}{\mathfrak{N}_{\alpha}(x)}

\\

& = & \displaystyle

\frac{1}{2} \left[ x {d \over dx} \left( \log \mathfrak{N}_{\alpha}(x) \right) +1 \right]

\end{array}

:

It can also be written as a ratio of two Wright functions,

: $\backslash begin\{array\}\{lcl\}$

\mu_\alpha(x) & = & \displaystyle

-\frac{1}{2} \frac{W_{-\alpha,-1}(-x^\alpha)}{\Gamma(\frac{1}{\alpha}+1) \, \mathfrak{N}_{\alpha}(x)}

\\

& = & \displaystyle

-\frac{1}{2} \frac{W_{-\alpha,-1}(-x^\alpha)}{W_{-\alpha,0}(-x^\alpha)}

\end{array}

When $\backslash alpha\; =\; 1/2$, this process is reduced to the modified CIR model where

$\backslash mu\_\{1/2\}(x)\; =\; \backslash frac\{1\}\{8\}\; (6-x)$.

This is the only special case where $\backslash mu\_\backslash alpha(x)$ is a straight line.

Likewise, if the asymptotic distribution is $V\_\{\backslash alpha\}(s)$ as $t\; \backslash rightarrow\; \backslash infty$,

the $\backslash mu\_\backslash alpha(x)$ solution, denoted as $\backslash mu(x;\; V\_\{\backslash alpha\})$ below, is

: $\backslash begin\{array\}\{lcl\}$

\mu(x; V_{\alpha}) & = & \displaystyle

- \frac{ W_{-\frac{\alpha}{2},-1}(-{(\sqrt{2} x)}^\alpha) }{ W_{-\frac{\alpha}{2},0}(-{(\sqrt{2} x)}^\alpha)}

-\frac{1}{2}

\end{array}

When $\backslash alpha\; =\; 1$, it is reduced to a quadratic polynomial:

$\backslash mu(x;V\_1)\; =\; 1\; -\; \backslash frac\{x^2\}\{2\}$.

By relaxing the rigid relation between the $\backslash sigma^2$ term and the $\backslash sigma$ term above,

the stable extension of the CIR model can be constructed as

: $dr\_t\; =\; a\; \backslash ,\; \backslash left[\; \backslash frac\{8b\}\{6\}\; \backslash ,$

\mu_{\alpha}\left( \frac{6}{b} r_t \right) \right] \, dt + \sigma \sqrt{r_t} \, dW,

which is reduced to the original CIR model

at $\backslash alpha\; =\; 1/2$:

$dr\_t\; =\; a\; \backslash left(\; b\; -\; r\_t\; \backslash right)\; dt\; +\; \backslash sigma\; \backslash sqrt\{r\_t\}\; \backslash ,\; dW$.

Hence, the parameter $a$ controls the mean-reverting speed,

the location parameter $b$ sets where the mean is, $\backslash sigma$ is the volatility parameter,

and $\backslash alpha$ is the shape parameter for the stable law.

By solving the Fokker-Planck equation, the solution for the PDF $p(x)$ at $r\_\backslash infty$ is

: $\backslash begin\{array\}\{lcl\}$

p(x) & \propto & \displaystyle

\exp \left[ \int^{x} \frac{dx}{x} \left(

2 D \, \mu_{\alpha}\left( \frac{6}{b} x \right) - 1

\right) \right]

, \text{ where } D = \frac{4ab}{3 \sigma^2}

\\

& = & \displaystyle

\mathfrak{N}_{\alpha}\left( \frac{6}{b} x \right) ^D \, x^{D-1}

\end{array}

To make sense of this solution, consider asymptotically for large $x$, $p(x)$'s tail is still in the form of a generalized gamma distribution, where in its $f(x;\; a\text{'},\; d,\; p)$ parametrization,

$p\; =\; \backslash frac\{\backslash alpha\}\{1-\backslash alpha\}$,

$a\text{'}\; =\; \backslash frac\{b\}\{6\}\; (D\backslash ,A(\backslash alpha))^\{-1/p\}$, and

$d\; =\; D\; \backslash left(\; 1\; +\; \backslash frac\{p\}\{2\}\; \backslash right)$.

It is reduced to the original CIR model

at $\backslash alpha\; =\; 1/2$ where $p(x)\; \backslash propto\; x^\{d-1\}e^\{-x/a\text{'}\}$

with $d\; =\; \backslash frac\{2ab\}\{\backslash sigma^2\}$ and

$A(\backslash alpha)\; =\; \backslash frac\{1\}\{4\}$; hence

$\backslash frac\{1\}\{a\text{'}\}\; =\; \backslash frac\{6\}\{b\}\; \backslash left(\backslash frac\{D\}\{4\}\backslash right)\; =\; \backslash frac\{2a\}\{\backslash sigma^2\}$.

From Section 4 of,{{cite arXiv|last1=Saxena|first1=R. K.|last2=Mathai|first2=A. M.|last3=Haubold|first3=H. J.|date=2009-09-01|title=Mittag-Leffler Functions and Their Applications|language=en|eprint=0909.0230|class=math.CA}} the inverse Laplace transform $H\_\backslash alpha(k)$ of the Mittag-Leffler function $E\_\backslash alpha(-x)$ is ($k>0$)

:$H\_\backslash alpha(k)=\; \backslash mathcal\{L\}^\{-1\}\backslash \{E\_\backslash alpha(-x)\backslash \}(k)$

= \frac{2}{\pi} \int_0^\infty E_{2\alpha}(-t^2) \cos(kt) \,dt.

On the other hand, the following relation was given by Pollard (1948),

:$H\_\backslash alpha(k)\; =$

\frac{1}{\alpha} \frac{1}{k^{1+1/\alpha}}

L_\alpha \left( \frac{1}{k^{1/\alpha}} \right).

Thus by $k=\backslash nu^\backslash alpha$, we obtain the relation between stable count distribution and Mittag-Leffter function:

:$\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)\; =$

\frac{\alpha^2 \nu^\alpha}{\Gamma \left(\frac{1}{\alpha}\right)}

H_\alpha(\nu^\alpha).

This relation can be verified quickly at $\backslash alpha=\backslash frac\{1\}\{2\}$ where $H\_\{\backslash frac\{1\}\{2\}\}(k)=\backslash frac\{1\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash ,e^\{-k^2/4\}$ and $k^2=\backslash nu$. This leads to the well-known quartic stable count result:

:$\backslash mathfrak\{N\}\_\{\backslash frac\{1\}\{2\}\}(\backslash nu)\; =$

\frac{\nu^{1/2}}{4\,\Gamma (2)} \times

\frac{1}{\sqrt{\pi}} \,e^{-\nu/4}

=

\frac{1}{4\,\sqrt{\pi}} \nu^{1/2}\,e^{-\nu/4}.

The ordinary Fokker-Planck equation (FPE) is $\backslash frac\{\backslash partial\; P\_1(x,t)\}\{\backslash partial\; t\}\; =\; K\_1\backslash ,\; \backslash tilde\{L\}\_\{FP\}\; P\_1(x,t)$

, where $\backslash tilde\{L\}\_\{FP\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; \backslash frac\{F(x)\}\{T\}\; +\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}$

is the Fokker-Planck space operator, $K\_1$

is the diffusion coefficient, $T$

is the temperature, and $F(x)$

is the external field. The time-fractional FPE introduces the additional fractional derivative $\backslash ,\_0D\_t^\{1-\backslash alpha\}$

such that $\backslash frac\{\backslash partial\; P\_\backslash alpha(x,t)\}\{\backslash partial\; t\}\; =\; K\_\backslash alpha\; \backslash ,\_0D\_t^\{1-\backslash alpha\}\; \backslash tilde\{L\}\_\{FP\}\; P\_\backslash alpha(x,t)$

, where $K\_\backslash alpha$

is the fractional diffusion coefficient.

Let $k=s/t^\backslash alpha$ in $H\_\backslash alpha(k)$, we obtain the kernel for the time-fractional FPE (Eq (16) of {{Cite journal|last=Barkai|first=E.|date=2001-03-29|title=Fractional Fokker-Planck equation, solution, and application|journal=Physical Review E|language=en|volume=63|issue=4|pages=046118|doi=10.1103/PhysRevE.63.046118|pmid=11308923|issn=1063-651X|bibcode=2001PhRvE..63d6118B|s2cid=18112355}})

:$n(s,t)\; =$

\frac{1}{\alpha} \frac{t}{s^{1+1/\alpha}}

L_\alpha \left( \frac{t}{s^{1/\alpha}} \right)

from which the fractional density $P\_\backslash alpha(x,t)$ can be calculated from an ordinary solution $P\_1(x,t)$ via

:$P\_\backslash alpha(x,t)\; =\; \backslash int\_0^\backslash infty$

n\left( \frac{s}{K},t\right) \,P_1(x,s) \,ds, \text{ where } K=\frac{K_\alpha}{K_1}.

Since $n(\backslash frac\{s\}\{K\},t)\backslash ,ds\; =\; \backslash Gamma\; \backslash left(\backslash frac\{1\}\{\backslash alpha\}+1\backslash right)$

\frac{1}{\nu}\,

\mathfrak{N}_\alpha(\nu; \theta=K^{1/\alpha}) \,d\nu

via change of variable $\backslash nu\; t\; =\; s^\{1/\backslash alpha\}$, the above integral becomes the product distribution with $\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu)$

, similar to the "lambda decomposition" concept, and scaling of time $t\; \backslash Rightarrow\; (\backslash nu\; t)^\backslash alpha$

:

:$P\_\backslash alpha(x,t)\; =$

\Gamma \left(\frac{1}{\alpha}+1\right)

\int_0^\infty

\frac{1}{\nu}\,

\mathfrak{N}_\alpha(\nu; \theta=K^{1/\alpha})

\,P_1(x,(\nu t)^\alpha) \,d\nu.

Here $\backslash mathfrak\{N\}\_\backslash alpha(\backslash nu;\; \backslash theta=K^\{1/\backslash alpha\})$

is interpreted as the distribution of impurity, expressed in the unit of $K^\{1/\backslash alpha\}$, that causes the anomalous diffusion.

• Lévy flight
• Lévy process
• Fractional calculus
• Anomalous diffusion
• Incomplete gamma function and Gamma distribution
• Poisson distribution

{{Reflist|30em}}

• R Package [https://cran.r-project.org/web/packages/stabledist/stabledist.pdf 'stabledist'] by Diethelm Wuertz, Martin Maechler and Rmetrics core team members. Computes stable density, probability, quantiles, and random numbers. Updated Sept. 12, 2016.

{{ProbDistributions|continuous-infinite}}

Category:Continuous distributions

Category:Probability distributions with non-finite variance

Category:Power laws

Category:Stability (probability)