asset price dynamics we will suppose that $S_t, t\ge 0,$follows the geometric Lévy proces, i.e. $S_t=e^{X_t}$ where $X_t,t\ge0,$ is a Lévy process. Then it is well known (cf. [@ConTan03],[@NBS19]) that the price of a contingent claim in the presence of jumps is given by a solution $V(t,S)$ of the following partial integro-differential equation: $$\begin{aligned}
\frac{\partial V}{\partial t} &+&\frac{\sigma^2}{2} S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S}-rV \nonumber
\\
&+&\int_{\mathbb{R}}\left[ V(t,Se^z)-V(t,S)-(e^z-1) S \frac{\partial V}{\partial S}(t,S) \right] \nu({\mathrm{d}}z)=0,
\label{PDE-S}
\\
&&V(T,S)=\Phi(S), \quad S>0, t\in[0,T).
\nonumber\end{aligned}$$ Here $\Phi$ is the pay-off diagram of a plain vanilla option. For example, $\Phi(S)=(S-K)^+$ for a call option, or $\Phi(S)=(K-S)^+$ for a put option where $K>0$ is the strike price. Here and after we shall denote by $a^+=\max(a,0)$ and $a^-=\min(a,0)$ the positive and negative parts of a real number $a$, respectively.

If we consider the following change of variables $V(t,S)=e^{-r\tau}u(\tau,x)$ where $\tau=T-t$, $x=\ln(\frac{S}{K})$ then we obtain the following PIDE for the function $u(\tau,x)$:

$$\begin{aligned}
\frac{\partial u}{\partial \tau}
&=&\frac{\sigma^2}{2} \frac{\partial^2 u}{\partial x^2} 
+ \left(r-\frac{1}{2}\sigma^2\right)\frac{\partial u}{\partial x}
\label{PDE-u-BS}
\\
&& + \int_{\mathbb{R}} \left[
u(\tau,x+z)-u(\tau,x)-\left(e^{z}-1\right)\frac{\partial u}{\partial x}(\tau,x)
\right]\nu({\mathrm{d}}z), 
\nonumber
\\
u(0,x)&=&\Phi(K e^{x}), \quad  x\in \mathbb{R}, \tau\in(0,T).
\nonumber\end{aligned}$$

Unfortunately, the initial condition $u(0,x)= \Phi(Ke^x)$ does not belong to the 