spaces $X^\gamma$ representing the fractional power spaces of the Lebesgue space $L^{p}({\mathbb{R}})$ with respect to the second order Laplacian operator. We follow the methodology of abstract semilinear parabolic equations developed by Henry in [@Henry1981]. First, we provide sufficient conditions guaranteeing existence and uniqueness of a solution to the PIDE (\[PDE-u\]) in Bessel potential spaces. In Section 4 we investigate qualitative properties of solutions to a PIDE of the Black-Scholes type arising in pricing derivative securities on underlying assets following Lévy processes. Section 5 is focused on application of the results for the nonlinear extension of the Black-Scholes PIDE for pricing American style of put options by the penalization method. Finally, in Section 6 we present results of a numerical solution to PIDE Variance Gamma and Merton’s models.

Preliminaries, definitions and motivation 
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A stochastic process $\{X_{t}, t\ge 0\}$ is called a Lévy stochastic process if its characteristic function has the following Lévy-Khintchine representation $\mathbb{E}\left[e^{i y X_{t}}\right]=e^{t\phi(y)}$ with $$\phi(y)=-\frac{\sigma^{2}}{2}y^{2}+i\omega y+\int_{-\infty}^{+\infty} \left(e^{i y z}-1-i y z1_{|z| \leq 1}\right) \nu  ({\mathrm{d}}z),$$ where $\sigma \geq 0$, $\omega \in \mathbb{R}$, and $\nu $ is a positive Radon measure on $\mathbb{R} \setminus \left\{0\right\}$ satisfying: $$\int_{\mathbb{R}} \min(z^2,1) \nu ({\mathrm{d}}z)<\infty, 
\label{condinttwo}$$ (cf. [@Sato99],[@ConTan03],[@App04]).

\[def-admissiblemeasure\] A Lévy measure $\nu$ 