e^{-A t}\Vert_{X^\gamma} = \Vert A^\gamma e^{-A t}\Vert_{X} \le M t^{-\gamma} e^{-at}$ of the norm of the semigroup $e^{-At}$ for any $t>0$.

As a direct consequence of Proposition \[semilinear\_general\_existence\_result\] and Lemma \[lemma-f\] we deduce the following result:

\[semilinear\_existence\_result\] Assume $\nu$ is an admissible activity Lévy measure with the shape parameters $\alpha, D^\pm$ and $\mu$ where $\alpha<3$ and either $\mu>0, D^\pm\in\mathbb{R}$, or $\mu=0, D^- + 1<0<D^+$. Assume $\gamma\ge 1/2$ and $\gamma>(\alpha-1)/2$. Suppose that the function $g(\tau,u)$ is Hölder continuous in the $\tau$ variable and Lipschitz continuous in the $u$ variable. Then for any $u_0\in X^\gamma$ and $T>0$ there exists the unique solution $u\in C([0,T],X^{\gamma})$ to PIDE (\[PDE-u\]).

The Black-Scholes PIDE model
============================

In this section, our purpose is to investigate properties of solutions to a PIDE generalizing the Black-Scholes model. An important definition concerning this generalization is definition of a Lévy measure of a given process $X_t$. The measure $\nu(A)$ of a Borel set $A\subseteq {\mathbb{R}}$ is defined by: $$\nu\left(A\right)=\mathbb{E}\left[\# \left\{t \in \left[0,1\right]:\Delta X_{t} \in A \right\}\right]=
\frac{1}{T}\mathbb{E}\left[\# \left\{t \in \left[0,T\right]:\Delta X_{t} \in A \right\}\right].
\label{eq:measuredef}$$ It gives the mean number, per unit of time, of jumps of $X_t, t\ge0,$ whose amplitude belongs to the set $A$ (see [@ConTan03]).

For the underlying 