Lévy models amenable to efficient calculations

In our previous publications (IJTAF 2019, Math. Finance 2020), we introduced a general class of SINH-regular processes and demonstrated that efficient numerical methods for the evaluation of the Wiener-Hopf factors and various probability distributions (prices of options of several types) in Lévy models can be developed using only a few general properties of the characteristic exponent $\psi$. Essentially all popular Lévy processes enjoy these properties. In the present paper, we define classes of Stieltjes-Lévy processes (SL-processes) as processes with completely monotone Lévy densities of positive and negative jumps, and signed Stieltjes-Lévy processes (sSL-processes) as processes with densities representable as differences of completely monotone densities. We demonstrate that 1) all crucial properties of $\psi$ are consequences of the representation $\psi (\xi )=(a_2^{+}{\xi }^2-i{a}_1^{+}\xi )ST({\text{cG}}_{+})(-i\xi )+(a_2^{-}{\xi }^2+i{a}_1^{-}\xi )ST({\text{cG}}_{-})(i\xi )+({\text{sg}}^2\mathrm{/}2){\xi }^2-i\mu \xi$, where $ST(\text{cG})$ is the Stieltjes transform of the (signed) Stieltjes measure $\text{cG}$ and $a_j^{\pm }\ge 0$; 2) essentially all popular processes other than Merton’s model and Meixner processes areSL-processes; 3) Meixner processes are sSL-processes; 4) under a natural symmetry condition, essentially all popular classes of Lévy processes are SL- or sSL-subordinated Brownian motion.