Predictable Variations in Stochastic Calculus

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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.