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A Feynman-Kac-type formula for Lévy processes with discontinuous killing rates. (English) Zbl 1355.60060

Author’s abstract: 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 as well as a broad class of pure jump processes such as generalized hyperbolic, multivariate normal inverse Gaussian, tempered stable, and\(\alpha \)-semistable Lévy processes.By virtue of our mild regularity assumptions as to the killing rate and the initial condition of the partial integro-differential equation, our results provide a rigorous basis for numerous applications, in financial mathematics and probability theory. We implement a Galerkin scheme to solve the corresponding pricing equation numerically and illustrate the effect of a killing rate.

MSC:

60G51 Processes with independent increments; Lévy processes
60H30 Applications of stochastic analysis (to PDEs, etc.)
35R09 Integro-partial differential equations
35S10 Initial value problems for PDEs with pseudodifferential operators
47G30 Pseudodifferential operators
47G20 Integro-differential operators
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
91G80 Financial applications of other theories
91G60 Numerical methods (including Monte Carlo methods)
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