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It is also proven that second moment of supremum of reflected fractional Brownian motion up to time $a$ is bounded above by $a^{2H}.$ Similar relations are obtained for the supremum of fractional Brownian motion with Hurst parameter greater than 1/2, and its hitting time to level $1.$ What is more, we obtain an u"},"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":"0910.5193","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-10-27T17:59:51Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"80d8a4f075aa5d3f00cbe8bb35559729fec2268ad8bec8231059030494ca497d","abstract_canon_sha256":"929da37f9ae32d65858b641bb59919cf385cc59c79fffd54b5b8d518ddbd4d31"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:48:59.394479Z","signature_b64":"lZ5lc8B6fL0fNGhhHVsOR637CLr6Xn8oBUXEjdDrGkTqeNFVGKKs6k53vo3/a+SST33sWb815pL/E6rt50n1Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b963239440ddea6f9d0be2bbf4f90aa8db1bb487dff01f98307bd9247a45fcf","last_reissued_at":"2026-05-18T03:48:59.393880Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:48:59.393880Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Results on the Supremum of Fractional Brownian Motion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Ceren Vardar","submitted_at":"2009-10-27T17:59:51Z","abstract_excerpt":"We show that the distribution of the square of the supremum of reflected fractional Brownian motion up to time a, with Hurst parameter-H greater than 1/2, is related to the distribution of its hitting time to level $1,$ using the self similarity property of fractional Brownian motion. 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