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

一维马尔可夫过程移动边界的边界交叉问题

• 批准号: 2443857
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2443857
• 关键词: Boundary; crossing; problems; dimensional; Markov

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.

该项目涉及各种马尔可夫过程的边界交叉问题（BCP）。目的是在随机时间找到明确和封闭式的解决方案，例如随机过程的第一个或最后一个通过时间分布，即遇到移动界限。研究此类问题既实用又是理论上的重要性。这些问题在许多科学领域都出现，例如数学物理，数学金融，神经病学和等。在布朗运动的情况下，这是一个经典的问题，并且可以针对线性，平方根或Quadratic等简单边界来得出明确的解决方案。图像方法使我们能够为更复杂的边界集获得此类结果，目标是将这些方法扩展到其他连续或跳跃马尔可夫过程（例如征费过程），并找到新的曲线家族，以便可以获得明确的结果。我们已经发表了一份题为“ Ornstein-uhlenbeck过程的边界交叉问题和功能转换”的手稿，如果我们研究了功能转换的两参数家族，并显示了其与Ornstein-Uhlenbeck（OU）类型的第一个通行时间（FPT）的联系。这样的打击时间问题引起了人们的极大兴趣，因为在许多应用程序中使用了OU过程来建模对象，例如金融中的利率或神经膜膜电压在神经科学中的演变。摘要和论文本身可以在此处找到（https://doi.org/10.48550/arxiv.2210.01658）。目前，我们对BCP感兴趣的跳跃过程感兴趣，例如频谱负征费过程（无积极跳跃的征费过程）或广义的OU过程，在这种过程中，我们没有驱动该过程的Brownian运动，而是具有频谱负征费过程。