Some properties of exponential integrals of Levy processes and examples
Hitoshi Kondo, Department of Mathematics, Keio University
Makoto Maejima, Department of Mathematics, Keio University
Ken-iti 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 non-selfdecomposable
but semi-selfdecomposable 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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