2008/04/03 by Antonis Papapantoleon, Papapantoleon, Antonis · 38 citations
Economics, Econometrics and Finance · Mathematics · #60E07 #60G44 #60G51 #91B28 (Primary) #Applied mathematics #Complex Systems and Time Series Analysis #Computer science #FOS: Economics and business #FOS: Mathematics #Financial Risk and Volatility Modeling #Girsanov theorem #Lévy process #Mathematical economics #Mathematical proof #Mathematics #Pricing of Securities (q-fin.PR) #Probability (math.PR) #Stochastic differential equation #Stochastic processes and financial applications #math.PR #msc:60E07 #msc:60G44 #msc:60G51 #msc:91B28 #q-fin.PR
paper · pdf · doi:10.48550/arxiv.0804.0482
published in arXiv (Cornell University) (Cornell University) · 50 pages, introductory lecture notes; typos corrected, change of format
openalex publication_date 2008/04/03 · arxiv created 2008/11/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
These lectures notes aim at introducing Lévy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of Lévy processes. We analyze a `toy' example of a Lévy process, viz. a Lévy jump-diffusion, which yet offers significant insight into the distributional and path structure of a Lévy process. Then, we present several important results about Lévy processes, such as infinite divisibility and the Lévy-Khintchine formula, the Lévy-Itô decomposition, the Itô formula for Lévy processes and Girsanov's transformation. Some (sketches of) proofs are presented, still the majority of proofs is omitted and the reader is referred to textbooks instead. In the second part, we turn our attention to the applications of Lévy processes in financial modeling and option pricing. We discuss how the price process of an asset can be modeled using Lévy processes and give a brief account of market incompleteness. Popular models in the literature are presented and revisited from the point of view of Lévy processes, and we also discuss three methods for pricing financial derivatives. Finally, some indicative evidence from applications to market data is presented.