is called an admissible activity Lévy measure if there exists a nonnegative measurable function $h$ such that $\nu({\mathrm{d}}z) = h(z) {\mathrm{d}}z$ such that $$0 \le  h(z)\le C_0 |z|^{-\alpha}\left(e^{D^{-}z}1_{z\geq 0}+e^{D^{+}z}1_{z< 0}\right)e^{-\mu z^{2}},
\label{growth_measure}$$ for any $z\in{\mathbb{R}}$ and the shape parameters $\alpha\geq 0$, $D^{\pm}\in\mathbb{R}$ and $\mu\geq 0$. Here $C_0>0$ is a constant.

The condition (\[condinttwo\]) is satisfied for any measure $\nu$ belonging to the class of admissible activity Lévy measures with shape parameters $0\le \alpha<3$, and either $\mu>0$ or $\mu=0$ and $D^-<0<D^+$.

Examples of admissible Lévy measures arising in the financial modelling
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The class of admissible activity Lévy measures includes various measures often used in financial modelling of underlying stock dynamics with jumps. For example, in the context of financial modelling the first jump-diffusion model was proposed by Merton in [@Merton76]. Its Lévy measure is given by: $$\nu({\mathrm{d}}z)=\lambda \frac{1}{\delta\sqrt{2\pi}}e^{-\frac{(z-m)^2}{2\delta^2}}{\mathrm{d}}z\,, 
\label{merton-density}$$ where $m\in{\mathbb{R}}, \lambda, \delta>0,$ are given parameters.

Another popular model is the so-called double exponential model introduced by Kou in [@Kou2002]. In this model, the Lévy measure $\nu$ is given by $$\nu ({\mathrm{d}}z)=\lambda \left( \theta \lambda^{+} e^{- \lambda^{+} z}1_{z\ge 0}+ (1-\theta) \lambda^{-} e^{\lambda^{-} z}1_{z<0}\right){\mathrm{d}}z,
\label{double-density}$$ where $\lambda>0$ is the intensity of jumps, $\theta$ is the probability of occurrence of positive jumps 