Real-valued self-similar Markov processes and their applications

随机过程是粒子在空间中随机运动时随时间演化的数学模型。有许多不同的随机过程家族，现在，当建立其他数学模型并在物理、生物、经济和工程中应用时，人们已经很好地理解了这些随机过程，并取得了不同程度的成功。在一些更常用的随机过程中，有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的当前位置，而不取决于它到目前为止的历史路径。这个建议的目的是构造新的马尔可夫随机过程族，它尊重一个基本的定义性质，即自相似。粗略地说，随机过程是自相似的，当在空间和时间上经过适当的重新缩放后，所产生的随机轨道是其自身的精确随机副本。这类新的自相似马尔可夫随机过程的重要性也将通过它们在许多不同的概率设置中的应用来探索。具体地说，我们将：(A)奠定数学基础，证明我们感兴趣的自相似马尔可夫过程的存在性。(B)参考马尔可夫过程的一般理论，探索它们的一些不寻常的性质。例如，我们将提供对我们的自相似过程族可能出现的奇怪现象的理解，因为它们的随机轨迹“从无穷大开始”。(C)看看“由自相似马尔可夫随机过程驱动”的随机微分方程式。前者可被认为是一族随机过程，其无穷小增量(或任意小规模随机运动)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的密切关系，称为马尔可夫可加L过程，得到关于后者的新结果，这些结果本身可用于其他应用。(E)利用所有上述知识并将其提供给一些具体的概率应用，称为最优停止问题。事实证明，后者在与金融建模有关的各种情景中具有突出的地位。

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.