## Electronic Communications in Probability

### 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

#### Article information

Source
Electron. Commun. Probab., Volume 11 (2006), paper no. 30, 291-303.

Dates
Accepted: 4 December 2006
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ecp/1465058873

Digital Object Identifier
doi:10.1214/ECP.v11-1232

Mathematical Reviews number (MathSciNet)
MR2266719

Zentralblatt MATH identifier
1130.60060

Rights

#### Citation

Kondo, Hitoshi; Maejima, Makoto; Sato, Ken-iti. Some properties of exponential integrals of Levy processes and examples. Electron. Commun. Probab. 11 (2006), paper no. 30, 291--303. doi:10.1214/ECP.v11-1232. https://projecteuclid.org/euclid.ecp/1465058873

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