formula for the first hitting time.\
7. We analyze in detail a number of concrete examples of the Levy processes which are used in the financial mathematics (stable processes, the variance damped Levy processes, the variance gamma processes, the normal Gaussian process, the Meixner process, compound Poisson process.)\

Main notions
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Let us consider the Levy processes $X_{t}$ on $R$. If $P(X_{0}=0)=1$ then Levy-Khinchine formula gives (see\[4\],\[23\]) $$\mu(z,t)=E\{\mathrm{exp}[izX_{t}]\}=
\mathrm{exp}[-t\lambda(z)],\quad t{\geq}0,$$ where $$\lambda(z)=\frac{1}{2}Az^{2}-i{\gamma}z-\int_{-\infty}^{\infty}(e^{ixz}-1-ixz1_{D(x)})d\nu(x).$$ Here $A{\geq}0,\quad \gamma=\overline{\gamma},$ and $D=\{x:|x|{\leq}1\}$ is the segment \[-1,1\], $\nu(x)$ is a monotonically increasing function satisfying the conditions $$\int_{-\infty}^{\infty}\frac{x^{2}}{1+x^{2}}d\nu(x)<\infty.$$ By $P_{t}(x_{0},\Delta)$ we denote the probability $P(X_{t}{\in}\Delta)$ when $P(X_{0}=x_{0})=1$ and $\Delta{\in}R$. The transition operator is defined by the formula $$P_{t}f(x)=\int_{-\infty}^{\infty}P_{t}(x,dy)f(y).$$ Let $C_{0}$ be the Banach space of continuous functions $f(x) \quad
(-\infty<x<\infty)$ satisfying the condition $\mathrm{lim}f(x)=0,\quad |x|{\to}\infty$ with the norm $||f||=
\mathrm{sup}_{x}|f(x)|$. We denote by $C_{0}^{n}$ the set of $f(x){\in}C_{0}$ such that $f^{(k)}(x){\in}C_{0},\quad
(1{\leq}k{\leq}n).$ It is known that \[23\] $$P_{t}f{\in}C_{0},$$ if $f(x){\in}C_{0}.$\
Now we formulate the following important result (see \[4\],\[23\]) .\
**Theorem 1.1.***The family of the operators $P_{t}\quad
(t{\geq}0)$ defined by the Levy process $X_{t}$ is a strongly continuous semigroup on $C_{0}$ with the norm $||P_{t}||=1$. Let $L$ be its infinitesimal generator. Then* $$Lf=\frac{1}{2}A\frac {d^{2}f}{dx^{2}}+\gamma
\frac{df}{dx}+\int_{-\infty}^{\infty}(f(x+y)-f(x)-y\frac{df}{dx}1_{D(x)})d\nu(x),$$ *where* $f{\in}C_{0}^{2}$.

