Predictable Variations in Stochastic Calculus

随机微积分关注的是理解和分析随机过程，这是随机系统如何随时间演变的描述。当研究随机过程的变换时，传统的方法集中在过程水平的操纵（即，一个过程的价值）。本研究建议将重点转移到过程增量的操纵（即，过程水平的变化）。这两种观点在数学上是等价的，但对过程增量的关注往往简化了计算，并导致新的见解。该项目首先考虑离散采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，并且可能是随机的。随着采样频率的增加，该项目表明，一个数学上方便的表示转换过程中的原始过程出现。例如，作为一种特殊情况，如果增量被线性变换，极限对象（随着采样频率的增加和时间间隔的长度趋于零）是相对于原始随机过程的随机积分。通过关注增量而不是水平的转换，该项目的目标是获得比目前在literation.The项目的第二阶段中可用的转换过程的更一般的表示，开始了第一步，将这些见解应用到与经典随机微积分相关的领域。其中一个领域是粗糙路径领域，粗略地说，随机过程被确定性路径所取代;另一个领域是量子随机微积分，现在随机过程模拟量子系统随时间的行为。

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.