Predictable Variations in Stochastic Calculus

随机微积分涉及理解和分析随机过程，它描述了随机系统如何随着时间的推移而演变。在研究随机过程的变换时，传统的方法侧重于过程层次(即过程的值)的操纵。这项研究建议将重点转移到过程增量(即过程级别的变化)的操作上。这两个观点在数学上是等价的，但对过程增量的关注往往会简化计算并产生新的见解。该项目首先考虑离散抽样随机过程的增量的变换。这种转变可能取决于时间和历史，也可能是随机的。随着采样频率的增加，该项目表明，根据原始过程，出现了转换过程的数学方便表示。例如，作为特例，如果增量是线性变换的，则极限对象(随着采样频率的增加和时间间隔的长度变为零)是相对于原始随机过程的随机积分。然后，该项目继续研究更复杂的变换。通过关注增量而不是水平的转换，该项目旨在获得比目前文献中可用的转换过程更通用的表示。该项目的第二阶段启动了将这些洞察力应用于与经典随机微积分相关的领域的第一步。一个这样的领域是粗糙路径领域，在那里随机过程被松散地说是确定性路径所取代。另一个领域是量子随机微积分，现在的随机过程对量子系统随时间的行为进行建模。

Stochastic calculus is concerned with understanding and analysing stochastic processes, which are descriptions of how random systems evolve over time. When studying transformations of stochastic processes, the traditional approach focuses on the manipulation of process levels (i.e., the values of a process). This research proposal shifts the focus to manipulations of process increments (i.e., the changes in the process levels). These two viewpoints are mathematically equivalent but the focus on process increments often simplifies computations and leads to new insights.The project begins by considering a transformation of the increments of a discretely sampled stochastic process. The transformation may depend on time and history and may be random. As the sampling frequency increases, the project shows that a mathematically convenient representation of the transformed process in terms of the original process emerges. For example, as a special case, if the increments are linearly transformed, the limiting object (as the sampling frequency increases and the lengths of the time intervals go to zero) is a stochastic integral with respect to the original stochastic process.The project then continues by looking at more involved transformations. By focusing on transformations of increments rather than levels, the project aims to derive more general representations of transformed processes than currently available in the literature.The second phase of the project initiates the first steps towards applying these insights to fields related to classical stochastic calculus. One such area is the field of rough paths, where the stochastic process is replaced, loosely speaking, by a deterministic path. Another area is quantum stochastic calculus, where now stochastic processes model the behaviour of quantum systems over time.