arXiv 2003.020732020-03-04math.PR

关于Killed指数泛函定律

On the law of killed exponential functionals

Anita Behme, Alexander Lindner, Jana Reker

DOI：: 10.1214/21-EJP616

摘要：

对于两个独立的Lévy过程$ξ$和$η$，以及一个参数$q>0$独立于$ξ$和$η$的指数分布随机变量$τ$，杀死的指数泛函由$V_{q,ξ,η} := \int_0^τ\mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$给出.利用马尔可夫过程的平稳分布产生的killed指数泛函，计算了马尔可夫过程的无穷小生成元，并利用它导出了描述$V_{q,ξ,η}$定律的分布方程，以及在绝对连续情况下Lebesgue密度的函数方程.各种特殊情况和例子被认为是，产生更明确的信息的法律杀害指数功能，并说明所获得的方程的应用。将$q=0$解释为$τ=\infty$导致经典的指数函数$\int_0^\infty \mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$，允许扩展许多先前的结果以包括杀伤。

英文摘要：

For two independent Lévy processes $ξ$ and $η$ and an exponentially distributed random variable $τ$ with parameter $q>0$ that is independent of $ξ$ and $η$, the killed exponential functional is given by $V_{q,ξ,η} := \int_0^τ\mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$. With the killed exponential functional arising as the stationary distribution of a Markov process, we calculate the infinitesimal generator of the process and use it to derive different distributional equations describing the law of $V_{q,ξ,η}$, as well as functional equations for its Lebesgue density in the absolutely continuous case. Various special cases and examples are considered, yielding more explicit information on the law of the killed exponential functional and illustrating the applications of the equations obtained. Interpreting the case $q=0$ as $τ=\infty$ leads to the classical exponential functional $\int_0^\infty \mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$, allowing to extend many previous results to include killing.

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英文摘要：

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