Zbigniew Michna Department of Mathematics Wroclaw University of
Economics Wroclaw, Poland

Applications of series representation for Levy processes''



We consider some applications of certain series representations for
Levy processes. We investigate two problems.  The first one concerns a
symmetric $\alpha$-stable Levy motion. We use a series representation
of the symmetric $\alpha$-stable Levy motion to condition on the
largest jump. As a component we get a Levy motion parametrised by
$x>0$ which has finite moments of all orders. This process in the
first expansion is a Brownian motion which gives that this Levy motion
can be approximated by Brownian motion for large $x$.

We show that this Levy motion converges to the symmetric
$\alpha$-stable Levy motion uniformly on compact sets with probability
one as $x\downarrow 0$. We also study integral of a non-random
function with respect to this L\'evy motion and derive the covariance
function of those integrals. A symmetric $\alpha$-stable random vector
is approximated with probability one by a random vector with
components having finite second moments.

The second one concerns asymptotic of the finite time ruin probability
for a gamma L\'evy process. We give the exact forms of the constants
$C_1$, $C_2$ and the function $g$ where
$C_1\leq\liminf_{u\rightarrow\infty}\Prob(\sup_{t\leq
T}(Z(t)-ct)>u)/g(u)\leq\limsup_{u\rightarrow\infty}\Prob(\sup_{t\leq
T}(Z(t)-ct)>u)/g(u)\leq C_2$ for any $T>0$ and $c>0$.