Abstract

In this article, we introduce generalized Mittag-Leffler Lévy (GMLL) process. GMLL distribution is represented as a general Lévy subordinator delayed by a gamma process. We show various properties of this new process like it’s corresponding Lévy density function and the so-called long-range dependence property. We provide also an explicit representation for the cumulative density function (CDF) for such process which can be an inspiration for construction of various versions of the GMLL processes. Moreover we establish Fokker-Planck type equation for its one dimensional probability density functions (PDF). Our construction will allow us to link GMLL process to the first passage time of the Lévy subordinator.

摘要

在本文中，我们介绍了广义 Mittag-Leffler Lévy (GMLL) 过程。GMLL 分布表示为由伽马过程延迟的一般 Lévy 从属子。我们展示了这个新过程的各种特性，例如它对应的 Lévy 密度函数和所谓的长程依赖特性。我们还为此类过程的累积密度函数 (CDF) 提供了一个明确的表示，这可以启发构建各种版本的 GMLL 过程。此外，我们为其一维概率密度函数（PDF）建立了Fokker-Planck类型方程。我们的构造将使我们能够将 GMLL 流程与 Lévy 下属的第一次通过时间联系起来。