  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.
