论文标题
关于被杀死的指数功能的定律
On the law of killed exponential functionals
论文作者
论文摘要
对于两个独立的lévy流程$ξ$和$η$以及一个指数分布的随机变量$τ$带有参数$ q> 0 $，独立于$ξ$和$η$，被杀死的指数功能由$ v_ {q，q，q，ξ，η}：= = = = \ int_0^°\ \ \ \ \} {e} \ Mathrm {D}η_s$。随着被杀死的指数功能作为马尔可夫过程的固定分布，我们计算了该过程的无限发电机，并使用它来得出不同的分布方程，描述了$ v_ {q，q，ξ，η} $的定律，以及在绝对持续情况下的lebesgue密度的功能方程。考虑了各种特殊情况和示例，从而获得有关被杀死的指数功能定律的更明确的信息，并说明了获得的方程式的应用。将情况解释为$ q = 0 $为$τ= \ infty $导致经典的指数函数$ \ int_0^\ intrmy \ mathrm {e}^{ - ξ_{ - ξ_{s-}} \，\ mathrm {d} \ {d} phist，允许扩大许多先前的结果。
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.
