论文标题
Ornstein-Uhlenbeck过程的边界越过问题和功能转换
Boundary crossing problems and functional transformations for Ornstein-Uhlenbeck processes
论文作者
论文摘要
我们对Ornstein-Uhlenbeck过程到时变阈值的第一个通过时间的定律感兴趣。我们表明，这个问题与该过程的第一个通过时间的定律有关，与时变边界的功能转换家族的成员。对于参数的特定值，这些转换出现在标准的Ornstein-uhlenbeck桥的实现中。我们提供了三个不同的证据。第一个是基于布朗运动的类似结果，第二个使用了所谓的高斯过程的概括，第三个依赖于Lie组对称方法。我们研究了这些转换的特性，并研究了用于构建它们的参与操作员的代数和分析特性。我们还表明，这些转换将Sturm-Liouville方程解决方案的解决方案映射到相关的非线性普通微分方程的解决方案空间中。最后，我们通过图像方法来解释我们的结果，并提供新的曲线示例，并具有明确的第一通道时间密度。
We are interested in the law of the first passage time of an Ornstein-Uhlenbeck process to time-varying thresholds. We show that this problem is connected to the laws of the first passage time of the process to members of a two-parameter family of functional transformations of a time-varying boundary. For specific values of the parameters, these transformations appear in a realisation of a standard Ornstein-Uhlenbeck bridge. We provide three different proofs of this connection. The first one is based on a similar result for Brownian motion, the second uses a generalisation of the so-called Gauss-Markov processes and the third relies on the Lie group symmetry method. We investigate the properties of these transformations and study the algebraic and analytical properties of an involution operator which is used in constructing them. We also show that these transformations map the space of solutions of Sturm-Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we interpret our results through the method of images and give new examples of curves with explicit first passage time densities.
