processes. Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of solutions in the Bessel potential space representing a fractional power space of the space of Lebesgue $p$-integrable functions with respect to the second order Laplace differential operator. We generalize known existence results for a wider class of Lévy measures having strong singular kernel with the third order of singularity. We also prove existence and uniqueness of solutions to the penalized PIDE representing approximation of the linear complementarity problem for a PIDE arising in pricing American style of options.

The paper is organized as follows. In Section 2 we recall typical examples of Lévy measures arising in the financial modelling of stochastic processes with random jumps. We introduce a notion of an admissible activity Lévy measure. We show that this class of Lévy measures includes jump-diffusion finite activity measures present in e.g. Merton’s or Kou’s double exponential models as well as infinite activity Lévy measure appearing in e.g. Variance Gamma, Normal Inverse Gaussian or the so-called CGMY models. Section 3 is devoted to the proof of the main result on existence and uniqueness of solution to the PIDE (\[PDE-u\]) in the framework of the Bessel potential 