## The Annals of Applied Probability

### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 1 (2016), 328-359.

Dates
Received: April 2014
Revised: December 2014
First available in Project Euclid: 5 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1452003241

Digital Object Identifier
doi:10.1214/14-AAP1093

Mathematical Reviews number (MathSciNet)
MR3449320

Zentralblatt MATH identifier
1339.60050

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 26C15: Rational functions [See also 14Pxx]

#### Citation

Hackmann, Daniel; Kuznetsov, Alexey. Approximating Lévy processes with completely monotone jumps. Ann. Appl. Probab. 26 (2016), no. 1, 328--359. doi:10.1214/14-AAP1093. https://projecteuclid.org/euclid.aoap/1452003241

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