The scale function holds significant importance within the fluctuation theory of Levy processes, particularly in addressing exit problems. However, its definition is established through the Laplace transform, thereby lacking explicit representations in general. This paper introduces a novel series representation for this scale function, employing Laguerre polynomials to construct a uniformly convergent approximate sequence. Additionally, we derive statistical inference based on specific discrete observations, presenting estimators of scale functions that are asymptotically normal.

翻译：尺度函数在Lévy过程的波动理论中具有重要地位，尤其适用于求解退出问题。然而，该函数通过Laplace变换定义，通常缺乏显式表示。本文提出一种新的尺度函数级数表示方法，利用Laguerre多项式构造一致收敛的逼近序列。此外，基于特定离散观测数据，我们推导了统计推断，并给出了渐近正态的尺度函数估计量。