论文标题
几何布朗运动，具有仿射漂移及其时间融合
Geometric Brownian motion with affine drift and its time-integral
论文作者
论文摘要
几何布朗运动的关节分布及其时间融合是在Yor（1992）的开创性论文中使用Lamperti的转换得出的，从而从修改的Bessel函数方面导致了明确的解决方案。在本文中，我们使用与HEUN微分方程相关的简单拉普拉斯变换方法重新审视了这一经典结果。我们将方法扩展到几何布朗尼运动，并以仿射漂移的形式扩展，并表明该过程的关节分布及其时间整合可以通过双重共鸣的HEUN方程来确定。此外，该过程的关节拉普拉斯变换及其时间整合源来自溶液的渐近学。此外，我们通过在定价亚洲选项中使用结果的结果来提供一个应用程序。数值结果显示了这种新方法的准确性和效率。
The joint distribution of a geometric Brownian motion and its time-integral was derived in a seminal paper by Yor (1992) using Lamperti's transformation, leading to explicit solutions in terms of modified Bessel functions. In this paper, we revisit this classic result using the simple Laplace transform approach in connection to the Heun differential equation. We extend the methodology to the geometric Brownian motion with affine drift and show that the joint distribution of this process and its time-integral can be determined by a doubly-confluent Heun equation. Furthermore, the joint Laplace transform of the process and its time-integral is derived from the asymptotics of the solutions. In addition, we provide an application by using the results for the asymptotics of the double-confluent Heun equation in pricing Asian options. Numerical results show the accuracy and efficiency of this new method.
