Some properties of exponential integrals of Lévy processes and examples

Abstract

The improper stochastic integral $Z={\int }_0^{\infty -}\exp (-X_{s-})d{Y}_s$ is studied, where $\{(X_t,Y_t),t⩾0\}$ is a Lévy 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.

13 pages