Levy 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 Wiener-Hopf factors various probability distributions (prices options several types) in L\'evy models can be developed using only few properties characteristic exponent $\psi$. Essentially all popular enjoy these properties. present paper, define classes Stieltjes-L\'evy (SL-processes) as with completely monotone densities positive negative jumps, signed (sSL-processes) representable differences densities. We demonstrate 1) crucial $\psi$ are consequences representation $\psi(\xi)=(a^+_2\xi^2-ia^+_1\xi)ST(\cG_+)(-i\xi)+(a^-_2\xi^2+ia^-_1\xi)ST(\cG_-)(i\xi)+(\sg^2/2)\xi^2-i\mu\xi$, where $ST(\cG)$ is Stieltjes transform (signed) measure $\cG$ $a^\pm_j\ge 0$; 2) essentially other than Merton's model Meixner areSL-processes; 3) sSL-processes; 4) under natural symmetry condition, SL- or sSL-subordinated Brownian motion.