Mathematics Seminar
Date: Thursday, 06 June 2024
Time: 10:30 AM - 11:30 AM
Venue: Seminar Hall
Invariance of Stochastic integrals, and some related isomorphism
Purba Das
King's College London.
06-06-24

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Abstract

We study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We introduce the concept of quadratic roughness of a path along a partition sequence and show that for Hölder-continuous paths satisfying this roughness condition, the quadratic variation along balanced partitions is invariant with respect to the choice of the partition sequence. Using these results we derive a formulation of the pathwise Föllmer-Itô calculus which is invariant with respect to the partition sequence.

We further provide some isomorphisms on spaces which are relevant for stochastic integrals. We extend Ciesielski's isomorphism along a general sequence of partitions, and provide a characterization of Hölder regularity of a function in terms of its Schauer coefficients. We also provide an isomorphism between the space of $\alpha$-H\"older continuous functions with finite $p$-th variation and a subclass of infinite-dimensional matrix equipped with appropriate norms, in the spirit of Ciesielski.