Some properties of exponential integrals of Lévy processes and examples

Details Some properties of exponential integrals of Lévy processes and examples Some properties of exponential integrals of Lévy processes and examples

2006-06-04

arxiv.org/abs/math/0606084v1

Mathematics

Probability

13 pages

Scientific paper

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.

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