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跨越问题(BCP)。目标是找到随机时间的显式和闭合解，例如到达移动边界的随机过程的第一次或最后一次通过时间分布。研究这些问题具有重要的现实意义和理论意义。这类问题出现在许多科学领域，如数学物理、数学金融、神经学等。在布朗运动的情况下，这是一个经典问题，可以导出简单边界的显式解，如线性边界、平方根边界或二次边界。图像法使我们能够对一组更复杂的边界得到这样的结果，目的是将这种方法推广到其他连续或跳跃的马尔可夫过程(如Levy过程)，并找到新的曲线族，从而可以得到明确的结果。我们已经发表了题为《Ornstein-Uhlenbeck过程的边界跨越问题和泛函变换》的手稿，我们研究了一族双参数泛函变换，并证明了它与Ornstein-Uhlenbeck(OU)型过程的第一通过时间(FPT)之间的关系。这样的命中时间问题是非常有趣的，因为OU过程已经在许多应用中被用来建模对象，如金融中的利率或神经科学中神经元膜电压的演化。摘要和论文本身可在此处找到(https://doi.org/10.48550/arXiv.2210.01658).目前，我们对跳跃过程的BCP很感兴趣，例如谱负Levy过程(没有正跳跃的Levy过程)或广义OU过程，其中我们不是用布朗运动来驱动过程，而是谱负Levy过程。