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2006 Some properties of exponential integrals of Levy processes and examples
Hitoshi Kondo, Makoto Maejima, Ken-iti Sato
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Electron. Commun. Probab. 11: 291-303 (2006). DOI: 10.1214/ECP.v11-1232


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


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Hitoshi Kondo. Makoto Maejima. Ken-iti Sato. "Some properties of exponential integrals of Levy processes and examples." Electron. Commun. Probab. 11 291 - 303, 2006.


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

zbMATH: 1130.60060
MathSciNet: MR2266719
Digital Object Identifier: 10.1214/ECP.v11-1232

Primary: 60E07
Secondary: 60G51 , 60H05

Keywords: generalized Ornstein-Uhlenbeck process , L'evy process , selfdecomposability , semi-selfdecomposability , stochastic integral


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