Symmetric Tempered Stable Distributions

Description

Contains methods for simulation and for evaluating the pdf, cdf, and quantile functions for symmetric stable, symmetric classical tempered stable, and symmetric power tempered stable distributions.

Details

The DESCRIPTION file:

Index of help topics:

SymTS-package           Symmetric Tempered Stable Distributions
dCTS                    PDF of CTS Distribution
dPowTS                  PDF of PowTS Distribution
dSaS                    PDF of Symmetric Stable Distribution
pCTS                    CDF of CTS Distribution
pPowTS                  PDF of PowTS Distribution
pSaS                    CDF of Symmetric Stable Distribution
qCTS                    Quantile Function of CTS Distribution
qPowTS                  Quantile Function of PowTS Distribution
qSaS                    Quantile Function of Symmetric Stable
                        Distribution
rCTS                    Simulation from CTS Distribution
rPowTS                  Simulation from PowTS Distribution
rSaS                    Simulation from Symmetric Stable Distribution

Author(s)

Michael Grabchak <mgrabcha@uncc.edu> and Lijuan Cao <lcao2@uncc.edu>

Maintainer: Michael Grabchak <mgrabcha@uncc.edu>

References

M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.

S. T. Rachev, Y. S. Kim, M. L. Bianchi, and F. J. Fabozzi (2011). Financial Models with Levy Processes and Volatility Clustering. Wiley, Chichester.

J. Rosinski (2007). Tempering stable processes. Stochastic Processes and Their Applications, 117(6):677-707.

G. Samorodnitsky and M. Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, Boca Raton.

PDF of CTS Distribution

Description

Evaluates the pdf for the symmetric classical tempered stable distribution. When alpha=0 this is the symmetric variance gamma distribution.

Usage

dCTS(x, alpha, c = 1, ell = 1, mu = 0)

Arguments

Details

The integration is preformed using the QAWF method in the GSL library for C. For this distribution the Rosinski measure R(dx) = c*delta_ell(dx) + c*delta_(-ell)(dx), where delta is the delta function. The Levy measure is M(dx) = c*ell^(alpha) *e^(-x/ell)*x^(-1-alpha) dx. The characteristic function is, for alpha not equal 0,1:

f(t) = exp( 2*c*gamma(-alpha)*(1+ell^2 t^2)^(alpha/2)*(cos(alpha*atan(ell*t))-1)) *e^(i*t*mu),

for alpha = 1 it is

f(t) = (1+ell^2 t^2)^c*exp(-2*c*ell*t*atan(ell*t)) *e^(i*t*mu),

and for alpha=0 it is

f(t) = (1+t^2 ell^2)^(-c) *e^(i*t*mu).

Note

When alpha=0 and c<=.5, the pdf is unbounded. It is infinite at mu and the method returns Inf in that case. This does not affect pCTS, qCTS, or rCTS.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.

Examples

x = (-10:10)/10
dCTS(x,.5)

PDF of PowTS Distribution

Description

Evaluates the pdf for the symmetric power tempered stable distribution.

Usage

dPowTS(x, alpha, c = 1, ell = 1, mu = 0)

Arguments

Details

The integration is preformed using the QAWF method in the GSL library for C. For this distribution the Rosinski measure R(dx) = c*(alpha+ell+1)*(alpha+ell)*(1+|x|)^(-2-alpha-ell)(dx).

Note

We do not allow for the case alpha=0 and c<=.5*(1+ell), as, in this case, the pdf is unbounded. This does not affect pPowTS, qPowTS, or rPowTS.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.

Examples

x = (-10:10)/10
dPowTS(x,.5)

PDF of Symmetric Stable Distribution

Description

Evaluates the pdf for the symmetric alpha stable distribution. For alpha=1 this is the Cauchy distribution.

Usage

dSaS(x, alpha, c = 1, mu = 0)

Arguments

Details

The integration is preformed using the QAWF method in the GSL library for C. The characteristic function is

f(t) = e^(-c |t|^alpha) *e^(i*t*mu).

Author(s)

Michael Grabchak and Lijuan Cao

References

G. Samorodnitsky and M. Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, Boca Raton.

Examples

x = (-10:10)/10
dSaS(x,.5)

CDF of CTS Distribution

Description

Evaluates the cdf for the symmetric classical tempered stable distribution. When alpha=0 this is the symmetric variance gamma distribution.

Usage

pCTS(x, alpha, c = 1, ell = 1, mu = 0)

Arguments

Details

For details about this distribution see the the describtion of dCTS.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.

Examples

x = (-10:10)/10
pCTS(x,.5)

PDF of PowTS Distribution

Description

Evaluates the cdf for the symmetric power tempered stable distribution.

Usage

pPowTS(x, alpha, c = 1, ell = 1, mu = 0)

Arguments

Details

The integration is preformed using the QAWF method in the GSL library for C. For this distribution the Rosinski measure R(dx) = c*(alpha+ell+1)*(alpha+ell)*(1+|x|)^(-2-alpha-ell)(dx).

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.

Examples

x = (-10:10)/10
pPowTS(x,.5)

CDF of Symmetric Stable Distribution

Description

Evaluates the cdf for the symmetric alpha stable distribution. For alpha=1 this is the Cauchy distribution.

Usage

pSaS(x, alpha, c = 1, mu = 0)

Arguments

Details

The integration is preformed using the QAWF method in the GSL library for C. The characteristic function is

f(t) = e^(-c |t|^alpha) *e^(i*t*mu).

Author(s)

Michael Grabchak and Lijuan Cao

References

G. Samorodnitsky and M. Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, Boca Raton.

Examples

x = (-10:10)/10
pSaS(x,.5)

Quantile Function of CTS Distribution

Description

Evaluates the quantile function for the symmetric classical tempered stable distribution. When alpha=0 this is the symmetric variance gamma distribution.

Usage

qCTS(x, alpha, c = 1, ell = 1, mu = 0)

Arguments

Details

For details about this distribution see the the describtion of dCTS.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.

Examples

x = (1:9)/10
qCTS(x,.5)

Quantile Function of PowTS Distribution

Description

Evaluates the quantile function for the symmetric power tempered stable distribution.

Usage

qPowTS(x, alpha, c = 1, ell = 1, mu = 0)

Arguments

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.

Examples

x = (1:9)/10
qPowTS(x,.5)

Quantile Function of Symmetric Stable Distribution

Description

Evaluates the quantile function for the symmetric alpha stable distribution. For alpha=1 this is the Cauchy distribution.

Usage

qSaS(x, alpha, c = 1, mu = 0)

Arguments

Details

The characteristic function is

f(t) = e^(-c |t|^alpha) *e^(i*t*mu).

Author(s)

Michael Grabchak and Lijuan Cao

References

G. Samorodnitsky and M. Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, Boca Raton.

Examples

x = (1:9)/10
qSaS(x,.5)

Simulation from CTS Distribution

Description

Simulates from the symmetric classical tempered stable distribution. When alpha=0 this is the symmetric variance gamma distribution. The simulation is performed by numerically evaluating the quantile function.

Usage

rCTS(r, alpha, c = 1, ell = 1, mu = 0)

Arguments

Details

For details about this distribution see the the describtion of dCTS.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.

Examples

rCTS(10,.5)

Simulation from PowTS Distribution

Description

Simulates from the symmetric power tempered stable distribution. The simulation is performed by numerically evaluating the quantile function.

Usage

rPowTS(r, alpha, c = 1, ell = 1, mu = 0)

Arguments

Details

For this distribution the Rosinski measure R(dx) = c*(alpha+ell+1)*(alpha+ell)*(1+|x|)^(-2-alpha-ell)(dx).

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.

Examples

pPowTS(10,.5)

Simulation from Symmetric Stable Distribution

Description

Simulates from the symmetric alpha stable distribution. When alpha=1 this is the Cauchy distribution. The simulation is performed using a well-known approah. See for instance Proposition 1.7.1 in Samorodnitsky and Taqqu (1994).

Usage

rSaS(r, alpha, c = 1, mu = 0)

Arguments

Details

The characteristic function is

f(t) = e^(-c |t|^alpha)*e^(i*t*mu).

Author(s)

Michael Grabchak and Lijuan Cao

References

G. Samorodnitsky and M. Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, Boca Raton.

Examples

rSaS(10,.5)