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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 $$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1106.2302","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-06-12T10:53:28Z","cross_cats_sorted":[],"title_canon_sha256":"5bb6d8ed9b1c50a1c2318f3776fd8798c6e36a1c0989785da4dc56f0b7e016a3","abstract_canon_sha256":"7d79fba5b1a690323aa16dde50f04a2f436c04d0bd45b1492d06af08803af73f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:42.818460Z","signature_b64":"lG4pMYyZmvVo9hoJ+zj03SqIDVwvOh3H76eZMcnmA+v42PBsCzyDAMqpQWnUgC0s933q/UnlmCc5yLQLecvlCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c999e4020cc4675b70a08fe97ecbe3eb30463390143ba342963e2532b0351a9","last_reissued_at":"2026-05-18T04:19:42.817725Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:42.817725Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The generalized quadratic covariation for fractional Brownian motion with Hurst index less than 1/2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chao Chen, Junfeng Liu, Litan Yan","submitted_at":"2011-06-12T10:53:28Z","abstract_excerpt":"Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1/2$. In this paper we study the {\\it generalized quadratic covariation} $[f(B^H),B^H]^{(W)}$ defined by $$ [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. 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