kind (see [@ConTan03]). Since $K_{1}(x)\sim \sqrt{\pi/2} x^{-1/2} e^{-x}$ as $x\rightarrow \infty$ and $K_{1}(x)\sim x^{-1}$ as $x\rightarrow 0$ (see [@Abramowitz]) the measure $\nu$ is an admissible activity Lévy measure with the shape parameters $\mu=0$, $D^{+}=A+B>0$, $D^{-}=A-B<0$, $\alpha=2$. The Variance Gamma and NIG processes are special cases of generalized hyperbolic models.

Finally, the so-called CGMY distribution process introduced by Carr [*et al.*]{} in [@CGMY98],[@PD99] has four parameters $C,G,M$ and $Y$ with the the Lévy measure given by: $$\nu( {\mathrm{d}}z)= C_0 |z|^{-1-Y} \left( e^{G z} 1_{z<0}+e^{-M z} 1_{z>0}\right) {\mathrm{d}}z,
\label{CGMY-density}$$ where $C,G,M>0$ and $Y<2$. The parameter $C$ measures the overall level of activity. The parameters $G$ and $M$ are the left and right tail decay parameters, respectively. When $G=M$ the distribution is symmetric. The process has infinite activity and finite variation when $Y\in (0,1)$ and infinite variation for $Y\in[1,2)$. The measure $\nu$ is an admissible activity Lévy measure with the shape parameters $\mu=0, \alpha=1+Y<3$, and $D^{+}=G>0, D^{-}=-M<0$.

Existence and uniqueness results
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The goal of this section is to prove the main result of the paper regarding existence and uniqueness of a solution to the linear and nonlinear PIDE for a wide class of admissible activity Lévy measures. We can rewrite the PIDE 