2015/02/26 by Kathrin Glau, Glau, Kathrin
Economics, Econometrics and Finance · Mathematics · #35S10 #47G20 #47G30 #60-08 #60G51 #Computational Finance (q-fin.CP) #FOS: Economics and business #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:35S10 #msc:47G20 #msc:47G30 #msc:60-08 #msc:60G51 #q-fin.CP
paper · pdf · doi:10.48550/arxiv.1502.07531
Revision and a new section added: Numerical Example
openalex publication_date 2015/02/26 · arxiv created 2015/11/04 · arxiv updated 2015/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
The challenge to fruitfully merge state-of-the-art techniques from mathematical finance and numerical analysis has inspired researchers to develop fast deterministic option pricing methods. As a result, highly efficient algorithms to compute option prices in Lévy models by solving partial integro differential equations have been developed. In order to provide a solid mathematical foundation for these methods, we derive a Feynman-Kac representation of variational solutions to partial integro differential equations that characterize conditional expectations of functionals of killed time-inhomogeneous Lévy processes. We allow for a wide range of underlying stochastic processes, comprising processes with Brownian part, and a broad class of pure jump processes such as generalized hyperbolic, multivariate normal inverse Gaussian, tempered stable, and α-semi stable Lévy processes. By virtue of our mild regularity assumptions as to the killing rate and the initial condition of the partial differential equation, our results provide a rigorous basis for numerous applications, not only in financial mathematics but also in probability theory and relativistic quantum mechanics.