
Some properties of exponential integrals of Levy processes and examples

Hitoshi Kondo, Department of Mathematics, Keio University Makoto Maejima, Department of Mathematics, Keio University Keniti Sato, 
Abstract
The improper stochastic integral $Z=int_0^{infty}exp(X_{s})dY_s$ is
studied, where
${ (X_t ,Y_t) , t ges 0 }$
is a L'evy process on $R ^{1+d}$ with ${X_t }$ and ${Y_t }$
being $R$valued and $R ^d$valued, respectively. The condition for
existence and finiteness of $Z$ is given and then the law $law(Z)$ of $Z$
is considered. Some sufficient conditions for $law(Z)$ to be selfdecomposable
and some sufficient conditions for $law(Z)$ to be nonselfdecomposable
but semiselfdecomposable are given. Attention is paid to the case where
$d=1$, ${X_t}$ is a Poisson process, and ${X_t}$ and ${Y_t}$ are
independent. An example of $Z$ of type $G$ with selfdecomposable mixing
distribution is given

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Published on: December 4, 2006

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