{"paper":{"title":"An integral functional driven by fractional Brownian motion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Litan Yan, Xianye Yu, Xichao Sun","submitted_at":"2016-02-29T02:41:41Z","abstract_excerpt":"Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1$ and the weighted local time ${\\mathscr L}^H(\\cdot,t)$. In this paper, we consider the integral functional $$ {\\mathcal C}^H_t(a):=\\lim_{\\varepsilon\\downarrow 0}\\int_0^t1_{\\{|B^H_s-a|>\\varepsilon\\}}\\frac1{B^H_s-a}ds^{2H}\\equiv \\frac1{\\pi}{\\mathscr H}{\\mathscr L}^H(\\cdot,t)(a) $$ in $L^2(\\Omega)$ with $ a\\in {\\mathbb R}, t\\geq 0$ and ${\\mathscr H}$ denoting the Hilbert transform. We show that $$ {\\mathcal C}^H_t(a)=2\\left((B^H_t-a)\\log|B^H_t-a|-B^H_t+a\\log|a| -\\int_0^t\\log|B^H_s-a|\\delta B^H_s\\right) $$ for all $a\\in {\\mathbb R},"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.08801","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}