[GS_M_APP] Roughness in finance via Schauder Representation
ABSTRACT
This presentation will explain two distinct concepts for measuring the roughness of financial data. We first introduce the idea of the p-th variation of a real-valued continuous function along a general class of refining partition sequences. We demonstrate that the finiteness of the p-th variation of a given path is closely linked to the finiteness of the $l^p$-norm of the coefficients along a Schauder basis, analogous to how the Hölder exponent relates to the $l^\infty$-norm of the Schauder coefficients. This result establishes an isomorphism between the space of Hölder continuous functions with finite (generalized) p-th variation along a given partition sequence and a subclass of infinite-dimensional matrices, equipped with an appropriate norm, in the spirit of Ciesielski.
