Approximating Lévy processes with completely monotone jumps

Abstract

Lévy processes with completely monotone jumps appear frequently in various applications of probability. For example, all popular stock price models based on Lévy processes (such as the Variance Gamma, CGMY/KoBoL and Normal Inverse Gaussian) belong to this class. In this paper we continue the work started in [Int. J. Theor. Appl. Finance 13 (2010) 63-91, Quant. Finance 10 (2010) 629-644] and develop a simple yet very efficient method for approximating processes with completely monotone jumps by processes with hyperexponential jumps, the latter being the most convenient class for performing numerical computations. Our approach is based on connecting Lévy processes with completely monotone jumps with several areas of classical analysis, including Padé approximations, Gaussian quadrature and orthogonal polynomials.

Intra‐Horizon expected shortfall and risk structure in models with jumps

“…This holds for instance true when dealing with hyper‐exponential jump‐diffusion processes that have the particularity to allow for arbitrarily close approximations of Lévy processes with completely monotone jumps (cf. Jeannin & Pistorius, 2010, Cai & Kou, 2011; Hackmann & Kuznetsov, 2016). A discussion of this approach as well as of approximations of Lévy densities via hyper‐exponential jump densities is provided in the upcoming sections.…”

“…Barndorff‐Nielsen, 1997) as well as the whole class of stable and tempered stable processes (cf. Küchler & Tappe, 2013; Hackmann & Kuznetsov, 2016), containing the very popular Variance‐Gamma (VG) (cf. Madan & Seneta, 1990; Madan et al., 1998) and Carr‐Geman‐Madan‐Yor (CGMY) models (cf.…”

A new class of survival distribution for degradation processes subject to shocks

“…Business, economic and financial systems are characterized by both stochastic degradation and growth scenarios and these systems are also subject to a wide array of random shocks exactly as in our model. The mathematical advances described in (Hackmann 2015;Hackmann and Kuznetsov 2016) , for example, are indicative of this rich source of development potential. Important extensions that will come from future research by these and other investigators might include more efficient numerical methods for calculating tail probabilities of the survival function and other probabilistic quantities for shock-degradation processes.…”

Classical robots perturbed by Lévy processes: analysis and Lévy disturbance rejection methods

“…Thus as an approximation, it is worth modeling a jerk process as a superposition of compound Poisson processes. Some of the useful references containing these ideas are [16][17][18].…”