{"paper":{"title":"The generalized quadratic covariation for fractional Brownian motion with Hurst index less than 1/2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chao Chen, Junfeng Liu, Litan Yan","submitted_at":"2011-06-12T10:53:28Z","abstract_excerpt":"Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1/2$. In this paper we study the {\\it generalized quadratic covariation} $[f(B^H),B^H]^{(W)}$ defined by $$ [f(B^H),B^H]^{(W)}_t=\\lim_{\\epsilon\\downarrow 0}\\frac{2H}{\\epsilon^{2H}}\\int_0^t\\{f(B^{H}_{s+\\epsilon})-f(B^{H}_s)\\}(B^{H}_{s+\\epsilon}- B^{H}_s)s^{2H-1}ds, $$ where the limit is uniform in probability and $x\\mapsto f(x)$ is a deterministic function. We construct a Banach space ${\\mathscr H}$ of measurable functions such that the generalized quadratic covariation exists in $L^2$ and the Bouleau-Yor identity takes the form $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.2302","kind":"arxiv","version":2},"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"}