Some properties of exponential integrals of L\'evy processes and examples

The improper stochastic integral $Z={\int }_0^{\mathrm{\infty }-}\exp (-X_{s-})d{Y}_s$ is studied, where $\{(X_t,Y_t),t⩾0\}$ is a L'evy process on $\mathbb R ^{1+d}$with${X_t }$and${Y_t }$being$\mathbb R$-valuedand$\mathbb R ^d$-valued,respectively\mathrm{.}Theconditionforexistenceandfinitenessof$Z$isgivenandthenthelaw$\mathcal L(Z)$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.