A Wiener–Hopf based approach to numerical computations in fluctuation theory for Lévy processes

• Published in: Mathematical methods of operations research (Heidelberg, Germany) Vol. 78; no. 1; pp. 101 - 118
• Main Authors: Iseger, Peter Den, Gruntjes, Paul, Mandjes, Michel
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2013; Springer; Springer Nature B.V
• Subjects: Algorithms; Barriers; Business and Management; Calculus of Variations and Optimal Control; Optimization; Fourier transforms; Laplace transforms; Mathematics; Mathematics and Statistics; Operations research; Operations Research/Decision Theory; Original Article; Partial differential equations; Random variables; Stochastic models; Studies; Wiener Hopf factorization; Laplace transform; Lévy processes; Concave majorant; Laplace inversion
• ISSN: 1432-2994; 1432-5217
• DOI: 10.1007/s00186-013-0434-9

This paper focuses on numerical evaluation techniques related to fluctuation theory for Lévy processes; they can be applied in various domains, e.g., in finance in the pricing of so-called barrier options. More specifically, with denoting the running maximum of the Lévy process , the aim is to evaluate for . The starting point is the Wiener–Hopf factorization, which yields an expression for the transform of the running maximum at an exponential epoch (with the mean of this exponential random variable). This expression is first rewritten in a more convenient form, and then it is pointed out how to use Laplace inversion techniques to numerically evaluate In our experiments we rely on the efficient and accurate algorithm developed in den Iseger (Probab Eng Inf Sci 20:1–44, 2006 ). We illustrate the performance of the algorithm with various examples: Brownian motion (with drift), a compound Poisson process, and a jump diffusion process. In models with jumps, we are also able to compute the density of the first time a specific threshold is exceeded, jointly with the corresponding overshoot. The paper is concluded by pointing out how our algorithm can be used in order to analyze the Lévy process’ concave majorant.