Predictable Variations in Stochastic Calculus

随机微积分的可预测变化

• 批准号: EP/Y024524/1
• 负责人: Johannes Ruf
• 金额: $ 9.85万
• 依托单位: London School of Economics and Political Science
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY024524%2F1
• 关键词: Predictable; Variations; Stochastic; Calculus

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.

随机微积分涉及理解和分析随机过程，随机过程是对随机系统如何随时间演变的描述。在研究随机过程的变换时，传统方法侧重于过程级别（即过程的值）的操纵。该研究提案将重点转移到过程增量的操纵（即过程级别的变化）。这两种观点在数学上是等效的，但对过程增量的关注通常会简化计算并带来新的见解。该项目首先考虑离散采样随机过程的增量的变换。该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，根据原始过程，出现了转换过程的数学方便表示。例如，作为一种特殊情况，如果增量进行线性变换，则限制对象（随着采样频率增加且时间间隔长度变为零）是相对于原始随机过程的随机积分。继续关注更多涉及的转型。通过关注增量而不是水平的转换，该项目旨在获得比现有文献中更通用的转换过程表示。该项目的第二阶段启动了将这些见解应用到与经典随机微积分相关领域的第一步。其中一个领域是粗糙路径领域，宽泛地说，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，现在随机过程对量子系统随时间的行为进行建模。