The improper stochastic integral $Z=\int_0^{ \infty-}\exp(-X_{ s-})dY_s$ is studied, where $\{ (X_t, Y_t), t \geqslant 0 \}$ is a L\'evy process on $\mathbb R ^{ 1+d}$ with $\{ X_t \}$ and $\{ Y_t \}$ being $\mathbb R$-valued and $\mathbb R ^d$-valued, respectively. The condition for existence and finiteness of $Z$ is given and then the law $\mathcal L(Z)$ of $Z$ is considered. Some sufficient conditions for $\mathcal L(Z)$ to be selfdecomposable and some sufficient conditions for $\mathcal L(Z)$ 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.
Categories: math.PR