Stable count distribution

In probability theory, the stable count distribution is the conjugate prior of a one-sided stable distribution. This distribution was discovered by Stephen Lihn in his 2017 study of daily distributions of S&P 500 index and the VIX index.[1] The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.[2]

Stable count

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \alpha }$ ∈ (0, 1) — stability parameter ${\displaystyle \theta }$ ∈ (0, ∞) — scale parameter ${\displaystyle \nu _{0}}$ ∈ (−∞, ∞) — location parameter
Support	x ∈ R and x ∈ [${\displaystyle \nu _{0}}$, ∞)
PDF	${\displaystyle {\mathfrak {N}}_{\alpha }(x;\nu _{0},\theta )={\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}{\frac {1}{x-\nu _{0}}}L_{\alpha }({\frac {\theta }{x-\nu _{0}}})}$
CDF	integral form exists
Mean	${\displaystyle {\frac {\Gamma ({\frac {2}{\alpha }})}{\Gamma ({\frac {1}{\alpha }})}}}$
Median	not analytically expressible
Mode	not analytically expressible
Variance	${\displaystyle {\frac {\Gamma ({\frac {3}{\alpha }})}{2\Gamma ({\frac {1}{\alpha }})}}-\left[{\frac {\Gamma ({\frac {2}{\alpha }})}{\Gamma ({\frac {1}{\alpha }})}}\right]^{2}}$
Skewness	TBD
Ex. kurtosis	TBD
MGF	Fox-Wright representation exists

Of the three parameters defining the distribution, the stability parameter ${\displaystyle \alpha }$ is most important. Stable count distributions have ${\displaystyle 0<\alpha <1}$. The known analytical case of ${\displaystyle \alpha =1/2}$ is related to the VIX distribution (See Section 7 of [1]). All the moments are finite for the distribution.

Definition

Its standard distribution is defined as

${\displaystyle {\mathfrak {N}}_{\alpha }(\nu )={\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}{\frac {1}{\nu }}L_{\alpha }\left({\frac {1}{\nu }}\right),}$

where ${\displaystyle \nu >0}$ and ${\displaystyle 0<\alpha <1.}$

Its location-scale family is defined as

${\displaystyle {\mathfrak {N}}_{\alpha }(\nu$ ;\nu _{0},\theta )={\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}{\frac {1}{\nu -\nu _{0}}}L_{\alpha }\left({\frac {\theta }{\nu -\nu _{0}}}\right),}

where ${\displaystyle \nu >\nu _{0}}$, ${\displaystyle \theta >0}$, and ${\displaystyle 0<\alpha <1.}$

In the above expression, ${\displaystyle L_{\alpha }(x)}$is a one-sided stable distribution,[3] which is defined as following.

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

${\displaystyle L_{\alpha }(x)=f(x;\alpha ,1,\cos \left({\frac {\pi \alpha }{2}}\right)^{1/\alpha },0),}$

where ${\displaystyle 0<\alpha <1}$.

Consider the Lévy sum ${\displaystyle Y=\sum _{i=1}^{N}X_{i}}$ where ${\displaystyle X_{i}\sim L_{\alpha }(x)}$, then ${\displaystyle Y}$ has the density ${\textstyle {\frac {1}{\nu }}L_{\alpha }\left({\frac {x}{\nu }}\right)}$ where ${\textstyle \nu =N^{1/\alpha }}$. Set ${\displaystyle x=1}$, we arrive at ${\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}$ without the normalization constant.

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

Integral form

Based on the integral form of ${\displaystyle L_{\alpha }(x)}$ and ${\displaystyle q=\exp(-i\alpha \pi /2)}$, we have the integral form of ${\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}$ as

${\displaystyle {\begin{aligned}{\mathfrak {N}}_{\alpha }(\nu )&={\frac {2\alpha }{\pi \Gamma ({\frac {1}{\alpha }})}}\int _{0}^{\infty }e^{-{\text{Re}}(q)\,t^{\alpha }}{\frac {1}{\nu }}\sin({\frac {t}{\nu }})\sin(-{\text{Im}}(q)\,t^{\alpha })\,dt,{\text{ or }}\\&={\frac {2\alpha }{\pi \Gamma ({\frac {1}{\alpha }})}}\int _{0}^{\infty }e^{-{\text{Re}}(q)\,t^{\alpha }}{\frac {1}{\nu }}\cos({\frac {t}{\nu }})\cos({\text{Im}}(q)\,t^{\alpha })\,dt.\\\end{aligned}}}$

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

${\displaystyle {\begin{aligned}\Phi _{\alpha }(x)&={\frac {2\alpha }{\pi \Gamma ({\frac {1}{\alpha }})}}\int _{0}^{x}\int _{0}^{\infty }e^{-{\text{Re}}(q)\,t^{\alpha }}{\frac {1}{\nu }}\sin({\frac {t}{\nu }})\sin(-{\text{Im}}(q)\,t^{\alpha })\,dt\,d\nu \\&=1-{\frac {2\alpha }{\pi \Gamma ({\frac {1}{\alpha }})}}\int _{0}^{\infty }e^{-{\text{Re}}(q)\,t^{\alpha }}\sin(-{\text{Im}}(q)\,t^{\alpha })\,{\text{Si}}({\frac {t}{x}})\,dt,\\\end{aligned}}}$

where ${\displaystyle {\text{Si}}(x)=\int _{0}^{x}{\frac {\sin(x)}{x}}\,dx}$ is the sine integral function.

The Wright representation

In "Series representation", it is shown that the stable count distribution is a special case of the Wright function (See Section 4 of [4]):

${\displaystyle {\mathfrak {N}}_{\alpha }(\nu )={\frac {\alpha }{\Gamma \left({\frac {1}{\alpha }}\right)}}\phi _{-\alpha ,0}(-\nu ^{\alpha }),\,{\text{where}}\,\,\phi _{\lambda ,\mu }(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!\,\Gamma (\lambda n+\mu )}}.}$

This leads to the Hankel integral: (based on (1.4.3) of [5])

${\displaystyle {\mathfrak {N}}_{\alpha }(\nu )={\frac {\alpha }{\Gamma \left({\frac {1}{\alpha }}\right)}}{\frac {1}{2\pi i}}\int _{Ha}e^{t-(\nu t)^{\alpha }}\,dt,\,}$where Ha represents a Hankel contour.

Alternative derivation – lambda decomposition

Another approach to derive the stable count distribution is to use the Laplace transform of the one-sided stable distribution, (Section 2.4 of [1])

${\displaystyle \int _{0}^{\infty }e^{-zx}L_{\alpha }(x)\,dx=e^{-z^{\alpha }},}$where ${\displaystyle 0<\alpha <1}$.

Let ${\displaystyle x=1/\nu }$, 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,

${\displaystyle {\frac {1}{2}}{\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}e^{-|z|^{\alpha }}=\int _{0}^{\infty }{\frac {1}{\nu }}\left({\frac {1}{2}}e^{-|z|/\nu }\right)\left({\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}{\frac {1}{\nu }}L_{\alpha }\left({\frac {1}{\nu }}\right)\right)\,d\nu ,}$

where ${\displaystyle z\in {\mathsf {R}}}$.

This is called the "lambda decomposition" (See Section 4 of [1]) 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 ${\displaystyle \alpha >1}$. 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.

Stable Vol Distribution

A variant of the stable count distribution called the stable vol distribution ${\displaystyle V_{\alpha }(s)}$ can be derived from lambda decomposition too (See Section 6 of [4]). It expresses the Laplace transform of ${\displaystyle e^{-|z|^{\alpha }}}$ in terms of a Gaussian mixture such that

${\displaystyle {\frac {1}{2}}{\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}e^{-|z|^{\alpha }}={\frac {1}{2}}{\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}e^{-(z^{2})^{\alpha /2}}=\int _{0}^{\infty }{\frac {1}{s}}\left({\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}(z/s)^{2}}\right)V_{\alpha }(s)\,ds,}$

where

${\displaystyle V_{\alpha }(s)={\frac {{\sqrt {2\pi }}\,\Gamma ({\frac {2}{\alpha }}+1)}{\Gamma ({\frac {1}{\alpha }}+1)}}\,{\mathfrak {N}}_{\frac {\alpha }{2}}(2s^{2}),0<\alpha \leq 2.}$

This transformation is named generalized Gauss transmutation since it generalizes the Gauss-Laplace transmutation, which is equivalent to ${\displaystyle V_{1}(s)=2{\sqrt {2\pi }}\,{\mathfrak {N}}_{\frac {1}{2}}(2s^{2})=s\,e^{-s^{2}/2}}$.

Asymptotic properties

For stable distribution family, it is essential to understand its asymptotic behaviors. From,[3] for small ${\displaystyle \nu }$,

${\displaystyle {\begin{aligned}{\mathfrak {N}}_{\alpha }(\nu )&\rightarrow B(\alpha )\,\nu ^{\alpha },{\text{ for }}\nu \rightarrow 0{\text{ and }}B(\alpha )>0.\\\end{aligned}}}$

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

For large ${\displaystyle \nu }$,

${\displaystyle {\begin{aligned}{\mathfrak {N}}_{\alpha }(\nu )&\rightarrow \nu ^{\frac {\alpha }{2(1-\alpha )}}e^{-A(\alpha )\,\nu ^{\frac {\alpha }{1-\alpha }}},{\text{ for }}\nu \rightarrow \infty {\text{ and }}A(\alpha )>0.\\\end{aligned}}}$

This shows that the tail of ${\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}$ decays exponentially at infinity. The larger ${\displaystyle \alpha }$ is, the stronger the decay.

Moments

The n-th moment ${\displaystyle m_{n}}$ of ${\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}$ is the ${\displaystyle -(n+1)}$-th moment of ${\displaystyle L_{\alpha }(x)}$. 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 [1])

${\displaystyle {\begin{aligned}m_{n}&=\int _{0}^{\infty }\nu ^{n}{\mathfrak {N}}_{\alpha }(\nu )d\nu ={\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}\int _{0}^{\infty }{\frac {1}{t^{n+1}}}L_{\alpha }(t)\,dt.\\\end{aligned}}}$

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

${\displaystyle {\begin{aligned}m_{n}&={\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}\int _{0}^{\infty }\nu ^{n}\phi _{-\alpha ,0}(-\nu ^{\alpha })\,d\nu \\&={\frac {\Gamma ({\frac {n+1}{\alpha }})}{\Gamma (n+1)\Gamma ({\frac {1}{\alpha }})}},\,n\geq -1.\\\end{aligned}}}$

where ${\displaystyle \int _{0}^{\infty }r^{\delta }\phi _{-\nu ,\mu }(-r)\,dr={\frac {\Gamma (\delta +1)}{\Gamma (\nu \delta +\nu +\mu )}},\,\delta >-1,0<\nu <1,\mu >0.}$(See (1.4.28) of [5])

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

${\displaystyle m_{1}={\frac {\Gamma ({\frac {2}{\alpha }})}{\Gamma ({\frac {1}{\alpha }})}}}$

The variance is

${\displaystyle \sigma ^{2}={\frac {\Gamma ({\frac {3}{\alpha }})}{2\Gamma ({\frac {1}{\alpha }})}}-\left[{\frac {\Gamma ({\frac {2}{\alpha }})}{\Gamma ({\frac {1}{\alpha }})}}\right]^{2}}$

Moment generating function

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

${\displaystyle {\begin{aligned}M_{\alpha }(s)&=\sum _{n=0}^{\infty }{\frac {m_{n}\,s^{n}}{n!}}={\frac {1}{\Gamma ({\frac {1}{\alpha }})}}\sum _{n=0}^{\infty }{\frac {\Gamma ({\frac {n+1}{\alpha }})\,s^{n}}{\Gamma (n+1)^{2}}}\\&={\frac {1}{\Gamma ({\frac {1}{\alpha }})}}{}_{1}\Psi _{1}\left[({\frac {1}{\alpha }},{\frac {1}{\alpha }});(1,1);s\right],\,\,{\text{or}}\\&={\frac {1}{\Gamma ({\frac {1}{\alpha }})}}H_{1,2}^{1,1}\left[-s{\bigl |}{\begin{matrix}(1-{\frac {1}{\alpha }},{\frac {1}{\alpha }})\\(0,1);(0,1)\end{matrix}}\right]\\\end{aligned}}}$

As a verification, at ${\displaystyle \alpha ={\frac {1}{2}}}$, ${\displaystyle M_{\frac {1}{2}}(s)=(1-4s)^{-{\frac {3}{2}}}}$ (see below) can be Taylor-expanded to ${\displaystyle {}_{1}\Psi _{1}\left[(2,2);(1,1);s\right]=\sum _{n=0}^{\infty }{\frac {\Gamma (2n+2)\,s^{n}}{\Gamma (n+1)^{2}}}}$ via ${\displaystyle \Gamma ({\frac {1}{2}}-n)={\sqrt {\pi }}{\frac {(-4)^{n}n!}{(2n)!}}}$.

Known analytical case – quartic stable count

When ${\displaystyle \alpha ={\frac {1}{2}}}$, ${\displaystyle L_{1/2}(x)}$ is the Lévy distribution which is an inverse gamma distribution. Thus ${\displaystyle {\mathfrak {N}}_{1/2}(\nu$ ;\nu _{0},\theta )} is a shifted gamma distribution of shape 3/2 and scale ${\displaystyle 4\theta }$,

${\displaystyle {\mathfrak {N}}_{\frac {1}{2}}(\nu$ ;\nu _{0},\theta )={\frac {1}{4{\sqrt {\pi }}\theta ^{3/2}}}(\nu -\nu _{0})^{1/2}e^{-(\nu -\nu _{0})/4\theta },}

where ${\displaystyle \nu >\nu _{0}}$, ${\displaystyle \theta >0}$.

Its mean is ${\displaystyle \nu _{0}+6\theta }$ and its standard deviation is ${\displaystyle {\sqrt {24}}\theta }$. This called "quartic stable count distribution". The word "quartic" comes from Lihn's former work on the lambda distribution[6] where ${\displaystyle \lambda =2/\alpha =4}$. At this setting, many facets of stable count distribution have elegant analytical solutions.

The p-th central moments are ${\displaystyle {\frac {2\Gamma (p+3/2)}{\Gamma (3/2)}}4^{p}\theta ^{p}}$. The CDF is ${\textstyle {\frac {2}{\sqrt {\pi }}}\gamma \left({\frac {3}{2}},{\frac {\nu -\nu _{0}}{4\theta }}\right)}$ where ${\displaystyle \gamma (s,x)}$ is the lower incomplete gamma function. And the MGF is ${\displaystyle M_{\frac {1}{2}}(s)=e^{s\nu _{0}}(1-4s\theta )^{-{\frac {3}{2}}}}$. (See Section 3 of [1])

Special case when α → 1

As ${\displaystyle \alpha }$ becomes larger, the peak of the distribution becomes sharper. A special case of ${\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}$ is when ${\displaystyle \alpha \rightarrow 1}$. The distribution behaves like a Dirac delta function,

${\displaystyle {\mathfrak {N}}_{\alpha \rightarrow 1}(\nu )\rightarrow \delta (\nu -1),}$

where ${\displaystyle \delta (x)={\begin{cases}\infty ,&{\text{if }}x=0\\0,&{\text{if }}x\neq 0\end{cases}}}$, and ${\displaystyle \int _{0_{-}}^{0_{+}}\delta (x)dx=1}$.

Series representation

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

${\displaystyle {\begin{aligned}{\mathfrak {N}}_{\alpha }(x)&={\frac {\alpha }{\pi \Gamma ({\frac {1}{\alpha }})}}\sum _{n=1}^{\infty }{\frac {-\sin(n(\alpha +1)\pi )}{n!}}{x}^{\alpha n}\Gamma (\alpha n+1)\\&={\frac {\alpha }{\pi \Gamma ({\frac {1}{\alpha }})}}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}\sin(n\alpha \pi )}{n!}}{x}^{\alpha n}\Gamma (\alpha n+1)\\\end{aligned}}}$.

This series representation has two interpretations:

• First, a similar form of this series was first given in Pollard (1948),[7] and in "Relation to Mittag-Leffler function", it is stated that ${\displaystyle {\mathfrak {N}}_{\alpha }(x)={\frac {\alpha ^{2}x^{\alpha }}{\Gamma \left({\frac {1}{\alpha }}\right)}}H_{\alpha }(x^{\alpha }),}$ where ${\displaystyle H_{\alpha }(k)}$ is the Laplace transform of the Mittag-Leffler function ${\displaystyle E_{\alpha }(-x)}$ .
• Secondly, this series is a special case of the Wright function ${\displaystyle \phi _{\lambda ,\mu }(z)}$: (See Section 1.4 of [5])

${\displaystyle {\begin{aligned}{\mathfrak {N}}_{\alpha }(x)&={\frac {\alpha }{\pi \Gamma ({\frac {1}{\alpha }})}}\sum _{n=1}^{\infty }{\frac {(-1)^{n}{x}^{\alpha n}}{n!}}\,\sin((\alpha n+1)\pi )\Gamma (\alpha n+1)\\&={\frac {\alpha }{\Gamma \left({\frac {1}{\alpha }}\right)}}\phi _{-\alpha ,0}(-x^{\alpha }),\,{\text{where}}\,\,\phi _{\lambda ,\mu }(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!\,\Gamma (\lambda n+\mu )}},\lambda >-1.\\\end{aligned}}}$

The proof is obtained by the reflection formula of the Gamma function: ${\displaystyle \sin((\alpha n+1)\pi )\Gamma (\alpha n+1)=\pi /\Gamma (-\alpha n)}$, which admits the mapping: ${\displaystyle \lambda =-\alpha ,\mu =0,z=-x^{\alpha }}$ in ${\displaystyle \phi _{\lambda ,\mu }(z)}$. The Wright representation leads to analytical solutions for many statistical properties of the stable count distribution and establish another connection to fractional calculus.

Applications

Stable count distribution can represent the daily distribution of VIX quite well. It is hypothesized that VIX is distributed like ${\displaystyle {\mathfrak {N}}_{\frac {1}{2}}(\nu$ ;\nu _{0},\theta )} with ${\displaystyle \nu _{0}=10.4}$ and ${\displaystyle \theta =1.6}$ (See Section 7 of [1]). Thus the stable count distribution is the first-order marginal distribution of a volatility process. In this context, ${\displaystyle \nu _{0}}$ 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:

VIX daily distribution and fit to stable count

One form of mean-reverting SDE for ${\displaystyle {\mathfrak {N}}_{\frac {1}{2}}(\nu$ ;\nu _{0},\theta )} is based on a modified Cox–Ingersoll–Ross (CIR) model. Assume ${\displaystyle S_{t}}$ is the volatility process, we have

${\displaystyle dS_{t}={\frac {\sigma ^{2}}{8\theta }}(6\theta +\nu _{0}-S_{t})\,dt+\sigma {\sqrt {S_{t}-\nu _{0}}}\,dW,}$

where ${\displaystyle \sigma }$ is the so-called "vol of vol". The "vol of vol" for VIX is called VVIX, which has a typical value of about 85.[8]

This SDE is analytically tractable and satisfies the Feller condition, thus ${\displaystyle S_{t}}$ would never go below ${\displaystyle \nu _{0}}$. But there is an subtle issue between theory and practice. There has been about 0.6% probability that VIX did go below ${\displaystyle \nu _{0}}$. This is called "spillover". To address it, one can replace the square root term with ${\displaystyle {\sqrt {\max(S_{t}-\nu _{0},\delta \nu _{0})}}}$, where ${\displaystyle \delta \nu _{0}\approx 0.01\,\nu _{0}}$ provides a small leakage channel for ${\displaystyle S_{t}}$ to drift slightly below ${\displaystyle \nu _{0}}$.

Extremely low VIX reading indicates a very complacent market. Thus the spillover condition, ${\displaystyle S_{t}<\nu _{0}}$, carries a certain significance - When it occurs, it usually indicates the calm before the storm in the business cycle.

Fractional calculus

Relation to Mittag-Leffler function

From Section 4 of,[9] the inverse Laplace transform ${\displaystyle H_{\alpha }(k)}$ of the Mittag-Leffler function ${\displaystyle E_{\alpha }(-x)}$ is (${\displaystyle k>0}$)

${\displaystyle H_{\alpha }(k)={\mathcal {L}}^{-1}\{E_{\alpha }(-x)\}(k)={\frac {2}{\pi }}\int _{0}^{\infty }E_{2\alpha }(-t^{2})\cos(kt)\,dt.}$

On the other hand, the following relation was given by Pollard (1948),[7]

${\displaystyle H_{\alpha }(k)={\frac {1}{\alpha }}{\frac {1}{k^{1+1/\alpha }}}L_{\alpha }\left({\frac {1}{k^{1/\alpha }}}\right).}$

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

${\displaystyle {\mathfrak {N}}_{\alpha }(\nu )={\frac {\alpha ^{2}\nu ^{\alpha }}{\Gamma \left({\frac {1}{\alpha }}\right)}}H_{\alpha }(\nu ^{\alpha }).}$

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

${\displaystyle {\mathfrak {N}}_{\frac {1}{2}}(\nu )={\frac {\nu ^{1/2}}{4\,\Gamma (2)}}\times {\frac {1}{\sqrt {\pi }}}\,e^{-\nu /4}={\frac {1}{4\,{\sqrt {\pi }}}}\nu ^{1/2}\,e^{-\nu /4}.}$

Relation to time-fractional Fokker-Planck equation

The ordinary Fokker-Planck equation (FPE) is ${\displaystyle {\frac {\partial P_{1}(x,t)}{\partial t}}=K_{1}\,{\tilde {L}}_{FP}P_{1}(x,t)}$, where ${\displaystyle {\tilde {L}}_{FP}={\frac {\partial }{\partial x}}{\frac {F(x)}{T}}+{\frac {\partial ^{2}}{\partial x^{2}}}}$ is the Fokker-Planck space operator, ${\displaystyle K_{1}}$ is the diffusion coefficient, ${\displaystyle T}$ is the temperature, and ${\displaystyle F(x)}$ is the external field. The time-fractional FPE introduces the additional fractional derivative ${\displaystyle \,_{0}D_{t}^{1-\alpha }}$ such that ${\displaystyle {\frac {\partial P_{\alpha }(x,t)}{\partial t}}=K_{\alpha }\,_{0}D_{t}^{1-\alpha }{\tilde {L}}_{FP}P_{\alpha }(x,t)}$, where ${\displaystyle K_{\alpha }}$ is the fractional diffusion coefficient.

Let ${\displaystyle k=s/t^{\alpha }}$ in ${\displaystyle H_{\alpha }(k)}$, we obtain the kernel for the time-fractional FPE (Eq (16) of [10])

${\displaystyle n(s,t)={\frac {1}{\alpha }}{\frac {t}{s^{1+1/\alpha }}}L_{\alpha }\left({\frac {t}{s^{1/\alpha }}}\right)}$

from which the fractional density ${\displaystyle P_{\alpha }(x,t)}$ can be calculated from an ordinary solution ${\displaystyle P_{1}(x,t)}$ via

${\displaystyle P_{\alpha }(x,t)=\int _{0}^{\infty }n\left({\frac {s}{K}},t\right)\,P_{1}(x,s)\,ds,{\text{ where }}K={\frac {K_{\alpha }}{K_{1}}}.}$

Since ${\displaystyle n({\frac {s}{K}},t)\,ds=\Gamma \left({\frac {1}{\alpha }}\right){\frac {1}{\alpha \nu }}\,{\mathfrak {N}}_{\alpha }(\nu$ ;\theta =K^{1/\alpha })\,d\nu } via change of variable ${\displaystyle \nu t=s^{1/\alpha }}$, the above integral becomes the product distribution with ${\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}$, similar to the "lambda decomposition" concept, and scaling of time ${\displaystyle t\Rightarrow (\nu t)^{\alpha }}$:

${\displaystyle P_{\alpha }(x,t)={\frac {\Gamma \left({\frac {1}{\alpha }}\right)}{\alpha }}\int _{0}^{\infty }{\frac {1}{\nu }}\,{\mathfrak {N}}_{\alpha }(\nu$ ;\theta =K^{1/\alpha })\,P_{1}(x,(\nu t)^{\alpha })\,d\nu .}

Here ${\displaystyle {\mathfrak {N}}_{\alpha }(\nu$ ;\theta =K^{1/\alpha })} is interpreted as the distribution of impurity, expressed in the unit of ${\displaystyle K^{1/\alpha }}$, that causes the anomalous diffusion.

See also

• Lévy flight
• Lévy process
• Fractional calculus
• Anomalous diffusion

References

1. Lihn, Stephen (2017). "A Theory of Asset Return and Volatility Under Stable Law and Stable Lambda Distribution". SSRN 3046732.
2. Paul Lévy, Calcul des probabilités 1925
3. Penson, K. A.; Górska, K. (2010-11-17). "Exact and Explicit Probability Densities for One-Sided Lévy Stable Distributions". Physical Review Letters. 105 (21): 210604. arXiv:1007.0193. Bibcode:2010PhRvL.105u0604P. doi:10.1103/PhysRevLett.105.210604. PMID 21231282. S2CID 27497684.
4. Lihn, Stephen (2020). "Stable Count Distribution for the Volatility Indices and Space-Time Generalized Stable Characteristic Function". SSRN 3659383.
5. Mathai, A.M.; Haubold, H.J. (2017). Fractional and Multivariable Calculus. Springer Optimization and Its Applications. 122. Cham: Springer International Publishing. doi:10.1007/978-3-319-59993-9. ISBN 9783319599922.
6. Lihn, Stephen H. T. (2017-01-26). "From Volatility Smile to Risk Neutral Probability and Closed Form Solution of Local Volatility Function". Rochester, NY. doi:10.2139/ssrn.2906522. S2CID 157746678. SSRN 2906522. Cite journal requires |journal= (help)
7. Pollard, Harry (1948-12-01). "The completely monotonic character of the Mittag-Leffler function $E_a \left( { - x} \right)$". Bulletin of the American Mathematical Society. 54 (12): 1115–1117. doi:10.1090/S0002-9904-1948-09132-7. ISSN 0002-9904.
8. "VVIX white paper". www.cboe.com. Retrieved 2019-08-09.
9. Saxena, R. K.; Mathai, A. M.; Haubold, H. J. (2009-09-01). "Mittag-Leffler Functions and Their Applications". arXiv:0909.0230 [math.CA].
10. Barkai, E. (2001-03-29). "Fractional Fokker-Planck equation, solution, and application". Physical Review E. 63 (4): 046118. Bibcode:2001PhRvE..63d6118B. doi:10.1103/PhysRevE.63.046118. ISSN 1063-651X. PMID 11308923. S2CID 18112355.

External links

• R Package 'stabledist' by Diethelm Wuertz, Martin Maechler and Rmetrics core team members. Computes stable density, probability, quantiles, and random numbers. Updated Sept. 12, 2016.