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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.05941 |
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| _version_ | 1866910109172498432 |
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| author | Das, Purba Kim, Donghan Lim, Fang Rui |
| author_facet | Das, Purba Kim, Donghan Lim, Fang Rui |
| contents | We study pathwise $p$-th variation of continuous paths on a compact interval along a fixed partition sequence. Although the class of continuous paths with finite $p$-th variation is generally not linear, we develop a coefficient-based approach via Faber-Schauder expansions that, for any $p>1$, enables the construction of paths with prescribed $p$-th variation while preserving useful linear structures and Hölder regularity. We first construct continuous paths with linear $p$-th variation from suitable conditions on their Faber-Schauder coefficients. We then prescribe nonlinear $p$-th variation through a multiplicative transformation and show that, whenever nonempty, the class of Hölder continuous paths with a given $p$-th variation is dense in $C([0,1])$. Next, we introduce a transport procedure that turns a Banach subspace of continuous functions into a Banach subspace of paths with explicitly controlled $p$-th variation. We also prove stability of the associated pathwise Föllmer-Itô map on these transported subspaces. Finally, via time-changes, we show that this constructive framework extends from $q$-adic partition sequences to broader classes of dense $q$-refining partition sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05941 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Banach spaces of continuous paths with finite $p$-th variation Das, Purba Kim, Donghan Lim, Fang Rui Probability 26A45, 41A30, 42C40, 46E15, 60G17 We study pathwise $p$-th variation of continuous paths on a compact interval along a fixed partition sequence. Although the class of continuous paths with finite $p$-th variation is generally not linear, we develop a coefficient-based approach via Faber-Schauder expansions that, for any $p>1$, enables the construction of paths with prescribed $p$-th variation while preserving useful linear structures and Hölder regularity. We first construct continuous paths with linear $p$-th variation from suitable conditions on their Faber-Schauder coefficients. We then prescribe nonlinear $p$-th variation through a multiplicative transformation and show that, whenever nonempty, the class of Hölder continuous paths with a given $p$-th variation is dense in $C([0,1])$. Next, we introduce a transport procedure that turns a Banach subspace of continuous functions into a Banach subspace of paths with explicitly controlled $p$-th variation. We also prove stability of the associated pathwise Föllmer-Itô map on these transported subspaces. Finally, via time-changes, we show that this constructive framework extends from $q$-adic partition sequences to broader classes of dense $q$-refining partition sequences. |
| title | Banach spaces of continuous paths with finite $p$-th variation |
| topic | Probability 26A45, 41A30, 42C40, 46E15, 60G17 |
| url | https://arxiv.org/abs/2604.05941 |