Boundary crossing problems for one-dimensional Markov processes to moving boundaries

该项目关注的是各种马尔可夫过程的边界交叉问题（BCP）。目标是找到显式和封闭形式的解决方案，随机时间，如第一次或最后一次通过时间分布的随机过程击中移动边界。研究这些问题具有重要的理论意义和实践意义。这样的问题出现在许多科学领域，如数学物理学，数学金融学，神经学等的布朗运动的情况下，这是一个经典的问题，并明确解决方案可以推导出简单的边界，如线性，平方根或二次。图像的方法使我们能够得到这样的结果，为一个更复杂的边界集，目标是将这种方法扩展到其他连续或跳跃马尔可夫过程（如Levy过程），并找到新的曲线族，这样就可以得到明确的结果。我们已经发表了一篇题为“边界交叉问题和函数变换的Ornstein-Uhlenbeck过程”的手稿，我们研究了一个两参数的函数变换族，并显示了它与Ornstein-Uhlenbeck（OUH）型过程的首次通过时间（FPT）的联系。这样的命中时间问题是非常感兴趣的，因为在许多应用中，已经使用了随机过程来建模对象，例如金融中的利率或神经科学中的神经元膜电压的演变。摘要和论文本身可以在这里找到（https：//doi.org/10.48550/arXiv.2210.01658）。目前，我们感兴趣的是跳跃过程的BCP，例如谱负Levy过程（没有正跳跃的Levy过程）或广义的BCPs过程，其中不是布朗运动驱动过程，而是谱负Levy过程。

The project is concerned with boundary crossing problems (BCPs) for various Markov processes. The goal is to find explicit and closed form solutions for random times such as first or last passage time distribution of stochastic processes hitting moving boundaries. Investigating such problems are both of practical and theoretical importance. Such problems arise in many fields of sciences such as mathematical physics, mathematical finance, neurology & etc. In the case of the Brownian motion, this is a classical problem, and explicit solutions can be derived for simple boundaries such as linear, square root or quadratic. The method of images enables us to derive such results for a more complicated set of boundaries and the goal is to extend such methods to other continuous or jump Markov processes (such as Levy processes) and find new family of curves such that explicit results can be obtained. We already published a manuscript titled "Boundary crossing problems and functional transformations for Ornstein-Uhlenbeck processes", were we investigated a two-parameter family of functional transformations and showed its connection to the first passage time (FPT) of Ornstein-Uhlenbeck(OU) type processes to time varying thresholds. Such a hitting time problem is of great interest, as the OU process has been used in many applications to model objects such as interest rates in finance or the evolution of the neuronal membrane voltages in neuroscience. The abstract and the paper itself can be found here (https://doi.org/10.48550/arXiv.2210.01658). Currently, we are interested in BCPs for jump process such as Spectrally negative Levy processes (a Levy process with no positive jumps) or generalized OU processes where instead of having a Brownian motion driving the process, we have a spectrally negative Levy process.