Virtual superclass for Bernstein functions with nonzero Lévy density

A virtual superclass for all Bernstein functions which can represented by a Lévy density (no drift or killing rate). That means that there exists a Lévy measure \(\nu\) such that $$ \psi(x) = \int_0^\infty (1 - e^{-ux}) \nu(du) , x > 0 . $$

Details

Evaluation of Bernstein functions with Lévy densities

For continuous Lévy densities, the values of the Bernstein function are calculated with stats::integrate() by using the representation $$ \psi(x) = \int_{0}^{\infty} (1 - \operatorname{e}^{-ux}) \nu(du), \quad x > 0 , $$ and the values of the iterated differences are calculated by using the representation $$ (-1)^{j-1} \Delta^{j} \psi(x) = \int_{0}^{\infty} \operatorname{e}^{-ux} (1 - \operatorname{e}^{-u})^j \nu(du) , \quad x > 0 . $$

For discrete Lévy densities \(\nu(du) = \sum_{i} y_i \delta_{u_i}(du)\), the values of the Bernstein function are calculated by using the representation $$ \psi(x) = \sum_{i} (1 - \operatorname{e}^{-u_i x}) y_i, \quad x > 0 , $$ and the values of the iterated differences are calculated by using the representation $$ (-1)^{j-1} \Delta^{j} \psi(x) = \sum_{i} \operatorname{e}^{-u_i x} (1 - \operatorname{e}^{-u_i})^j y_i, \quad x > 0 . $$

See also

getLevyDensity(), calcIterativeDifference(), calcShockArrivalIntensities(), calcExShockArrivalIntensities(), calcExShockSizeArrivalIntensities(), calcMDCMGeneratorMatrix(), rextmo(), rpextmo()

Other Bernstein function classes: AlphaStableBernsteinFunction-class, BernsteinFunction-class, CompleteBernsteinFunction-class, CompositeScaledBernsteinFunction-class, ConstantBernsteinFunction-class, ConvexCombinationOfBernsteinFunctions-class, ExponentialBernsteinFunction-class, GammaBernsteinFunction-class, InverseGaussianBernsteinFunction-class, LinearBernsteinFunction-class, ParetoBernsteinFunction-class, PoissonBernsteinFunction-class, ScaledBernsteinFunction-class, SumOfBernsteinFunctions-class

Other Virtual Bernstein function classes: BernsteinFunction-class, CompleteBernsteinFunction-class

Other Levy Bernstein function classes: AlphaStableBernsteinFunction-class, CompleteBernsteinFunction-class, ExponentialBernsteinFunction-class, GammaBernsteinFunction-class, InverseGaussianBernsteinFunction-class, ParetoBernsteinFunction-class, PoissonBernsteinFunction-class