Real-valued self-similar Markov processes and their applications

随机过程（英语：Random process）是一种数学模型，描述一个在空间中随机移动的粒子随时间的演化。有许多不同的家庭的随机过程，现在，很好地理解不同程度的成功时，建立其他数学模型与应用在物理学，生物学，经济学和工程学。其中一些更常用的随机过程是所谓的马尔可夫随机过程。对于这些过程，粒子在任何时刻的未来随机演化只取决于它当前的位置，而不取决于它迄今为止的历史路径。这项建议的目的是建立新的家庭的马尔可夫随机过程，从来没有处理过，尊重一个基本的定义属性，即自相似性。粗略地说，随机过程是自相似的，当在空间和时间上进行适当的重新缩放后，所产生的随机轨迹是其自身的精确随机副本。这个新的家庭的自相似马尔可夫随机过程的重要性，也将探讨通过他们的应用程序在一些不同的概率设置。具体来说，我们将：（a）奠定数学基础，表明存在的自相似马尔可夫过程，我们感兴趣的。(b)探索他们的一些不寻常的属性与参考马尔可夫过程的一般理论。例如，我们将提供一个奇怪的现象，可以发生在我们的自相似家庭的过程中，他们的随机轨迹“从无穷大开始”的理解。(c)看看随机微分方程，它“由自相似马尔可夫随机过程驱动”。前者可以被认为是一类随机过程，其无穷小增量（或任意小尺度的随机运动）由后者的无穷小增量决定。(d)利用自相似马尔可夫过程与其他随机过程族（称为马尔可夫加性L'evy过程）的密切关系，导出关于后者的新结果，这些结果本身可以用于其他应用。(e)把上面所有的知识都应用到一些具体的概率应用中，称为最优停止问题。后者已证明在与金融建模有关的各种情况下都很突出。

A stochastic process is a mathematical model for the evolution through time of a particle that moves randomly through space. There are many different families of stochastic processes that are, now, well understood with varying degrees of success when building other mathematical models with applications in physics, biology, economics and engineering. Amongst some of the more commonly used stochastic processes are so-called Markov stochastic processes. For these processes, the future random evolution of the particle at any moment in time depends only on its current position and not on its historical path to date. This proposal aims to construct new families of Markov stochastic processes, never dealt with before, which respect a fundamental defining property, namely self-similarity. Roughly speaking, a stochastic process is self-similar when, after an appropriate re-scaling in space and time, the resulting random trajectory is an exact stochastic copy of itself. The importance of this new family of self-similar Markov stochastic processes will also be explored through their application in a number of different probabilistic settings. Specifically we shall:(a) Lay down the mathematical foundations, showing the existence of the self-similar Markov processes that we are interested in. (b) Explore some of their unusual properties with reference to the general theory of Markov processes. For example, we shall provide an understanding of the strange phenomenon that can occur with our self-similar family of processes in that their random trajectory "starts from infinity".(c) Look at stochastic differential equations which are "driven by a self-similar Markov stochastic processes". The former can be considered as a family of stochastic processes whose infinitesimal increments (or arbitrarily small-scale random movements) are determined by the infinitesimal increments of the latter.(d) Take advantage of the intimate relationship of self-similar Markov processes with other families of stochastic processes, known as Markov additive L'evy processes, to derive new results concerning the latter, which themselves can be fed into other applications.(e) Take all of the above knowledge and feed it into some concrete probabilistic applications known as optimal stopping problems. The latter have proved to be of prominence in a variety of scenarios which are pertinent to financial modelling.