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

Peter Iseger, Paul Gruntjes and Michel Mandjes ()

Mathematical Methods of Operations Research, 2013, vol. 78, issue 1, 118 pages

Abstract: 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 $$\bar{X}_t:= \sup _{0\le s\le t} X_s$$ denoting the running maximum of the Lévy process $$X_t$$ , the aim is to evaluate $$\mathbb{P }(\bar{X}_t \in \mathrm{d}x)$$ for $$t,x>0$$ . The starting point is the Wiener–Hopf factorization, which yields an expression for the transform $$\mathbb E e^{-\alpha \bar{X}_{e(\vartheta )}}$$ of the running maximum at an exponential epoch (with $$\vartheta ^{-1}$$ 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 $$\mathbb{P }(\bar{X}_t\in \mathrm{d}x).$$ 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. Copyright Springer-Verlag Berlin Heidelberg 2013

Keywords: Wiener Hopf factorization; Laplace transform; Laplace inversion; Lévy processes; Concave majorant (search for similar items in EconPapers)
Date: 2013
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

Downloads: (external link)
http://hdl.handle.net/10.1007/s00186-013-0434-9 (text/html)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:mathme:v:78:y:2013:i:1:p:101-118

Ordering information: This journal article can be ordered from
http://www.springer.com/economics/journal/00186

DOI: 10.1007/s00186-013-0434-9

Access Statistics for this article

Mathematical Methods of Operations Research is currently edited by Oliver Stein

More articles in Mathematical Methods of Operations Research from Springer, Gesellschaft für Operations Research (GOR), Nederlands Genootschap voor Besliskunde (NGB)
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().