operator in ${\mathbb{R}}^n$ with the domain $D(A)\equiv W^{2,p}({\mathbb{R}}^n)\subset X\equiv L^p({\mathbb{R}}^n)$ then $X^\gamma$ is embedded in the Sobolev-Slobodecki space $W^{2\gamma,p}({\mathbb{R}}^n)$ consisting of all functions having $2\gamma$-fractional derivative belonging to the Lebesgue space $L^p({\mathbb{R}}^n)$ of $p$-integrable functions (cf. [@Henry1981]). In this paper, our goal is to prove existence and uniqueness of solutions to (\[PDE-u\]) for a general class of the so-called admissible activity Lévy measures $\nu$ satisfying suitable growth conditions at $\pm\infty$ and the origin.

A motivation for studying solutions of the PIDE (\[PDE-u\]) arises from financial modeling. In the last four decades, the Black-Scholes model and its various generalizations become popular in the financial industry because of their simplicity and possibility to price options by means of explicit analytic formulas. However, practical application of the classical linear Black-Scholes equation has serious drawbacks, e.g. evidence from the stock market indicating that this model is less realistic as it assumes that the market is liquid, complete and without transaction costs. Moreover, sample paths of a Brownian motion are continuous, but stock prices of a typical company usually suffer from sudden jumps on an intra-day scale, making the price trajectories discontinuous. In the classical Black-Scholes model, the logarithm of the price process has a 