volatility of the underlying asset process $\{S_t\}_{t\ge0}$, $r\ge 0$ is the risk-less interest rate and $\nu$ is a Lévy measure. A solution $V$ is subject to the terminal condition $V(S,T)=\Phi(S)$ where $\Phi$ represents the pay-off diagram of a plain vanilla option, i.e. $\Phi(S)= (S-K)^+$ for a call option, or $\Phi(S)= (K-S)^+$ for a put option, $K>0$ is the strike price.

In the case when the Lévy measure $\nu$ is defined through the Dirac function, i.e. $ \nu({\mathrm{d}}z) =\delta(z) {\mathrm{d}}z$ or $\nu\equiv0$ the aforementioned nonlocal PIDE reduces to the classical linear Black-Scholes linear PDE: $$\frac{\partial V}{\partial t} +\frac{\sigma^2}{2} S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S}-rV =0.$$

In the past years, existence results of PIDE (\[PDE-u\]) have been intensively studied in the literature. In [@BL82] A. Bensoussan and J.-L. Lions (see Theorem 3.3 and Theorem 8.1) and also M. G. Garroni and J. L. Menaldi (see [@GarrMenal2002]) investigated the existence and uniqueness of classical solutions for the case $\sigma>0$. In [@RMikuPraga93] Mikulevicius and Pragarauskas extended these results for the case $\sigma=0$. Furthermore, in [@RMikuPraga2014],[@RMikuPraga14] they investigated existence and uniqueness of classical solutions in Hölder and Sobolev spaces of the Cauchy problem to the partial-integro-differential equation of the order of 