Some properties of exponential integrals of Levy processes and examples

Abstract

The improper stochastic integral $Z= \int_0^{\infty-}\exp(-X_{s-})dY_s$ is studied, where ${ (X_t ,Y_t) , t \geq 0 }$ is a Lévy 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 ${\cal L}(Z)$ of $Z$ is considered. Some sufficient conditions for ${\cal L}(Z)$ to be selfdecomposable and some sufficient conditions for ${\cal L}(Z)$ 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