  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 L\'evy 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 L\'evy 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 $\alpha$-semi stable L\'evy
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.
