2022/07/05 by Boyarchenko, Svetlana, Levendorskiĭ, Sergei
#60-08 #6051 #60G52 #65C05 #91G20 #Computational Finance (q-fin.CP) #FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Probability (math.PR)
paper · doi:10.48550/arxiv.2207.02359
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 ψ. 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 ψ are consequences of the representation ψ(ξ)=(a+2ξ2-ia+1ξ)ST(\cG+)(-iξ)+(a-2ξ2+ia-1ξ)ST(\cG-)(iξ)+(\sg2/2)ξ2-iμξ, where ST(\cG) is the Stieltjes transform of the (signed) Stieltjes measure \cG and a^±j≥ 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.