Electronic Communications in Probability

Some properties of exponential integrals of Levy processes and examples

Hitoshi Kondo, Makoto Maejima, and Ken-iti Sato

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60G51: Processes with independent increments; Lévy processes 60H05: Stochastic integrals

Generalized Ornstein-Uhlenbeck process L'evy process selfdecomposability semi-selfdecomposability stochastic integral

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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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