A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates

Kathrin Glau ()
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Kathrin Glau: Technical University of Munich

Finance and Stochastics, 2016, vol. 20, issue 4, No 7, 1059 pages

Abstract: 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 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 α $\alpha$ -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 in probability theory. We implement a Galerkin scheme to solve the corresponding pricing equation numerically and illustrate the effect of a killing rate.

Keywords: Time-inhomogeneous Lévy process; Killing rate; Feynman–Kac representation; Weak solution; variational solution; Parabolic evolution equation; Partial integro-differential equation; Pseudo-differential equation; Nonlocal operator; Fractional Laplace operator; Sobolev–Slobodeckii spaces; Option pricing; Laplace transform of occupation time; Employee option; Galerkin method; 60G51; 65M06; 35S10; 47G20; 47G30 (search for similar items in EconPapers)
JEL-codes: C10 C63 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (5)

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DOI: 10.1007/s00780-016-0301-7

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