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Fractional Brownian motion in Brownian motion time (of index $H$), recently studied in \\cite{13}, is by definition the process $Z=X\\circ Y$. It is a continuous, non-Gaussian process with stationary increments, which is selfsimilar of index $H/2$. The main result of the present paper is an It\\^{o}'s type formula for $f(Z_t)$, when $f:\\R\\to\\R$ is smooth and $H\\in [1/6,1)$. When $H>1/6$, the change-of-variable formula we obtain is similar to that of the"},"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":"1312.0818","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-03T13:36:46Z","cross_cats_sorted":[],"title_canon_sha256":"401cf9c097b1f8ebfebe6606318088695a255477f6a0cda1a89b195f34d8ecfd","abstract_canon_sha256":"db86dae08eaca301d810d4dc437c3a01feff4e52912fc1491ba5f5d266d68090"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:36.640112Z","signature_b64":"VQ3Y8nM6y1sezeZCRB5HbT0yn2GTrHmSIPAFugJd+TPZR0Pljo0bjaMZgOPU6/p84wVamdx7Gl7juQQvEvGBCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"43d0bc133e6a0de2f431fcc5b63fd8ba2ff4d32ac9d10a22eb982accab095fc2","last_reissued_at":"2026-05-18T03:05:36.639152Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:36.639152Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An It\\^o's type formula for the fractional Brownian motion in Brownian time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ivan Nourdin (IECL), Raghid Zeineddine (IECL)","submitted_at":"2013-12-03T13:36:46Z","abstract_excerpt":"Let $X$ be a (two-sided) fractional Brownian motion of Hurst parameter $H\\in (0,1)$ and let $Y$ be a standard Brownian motion independent of $X$. 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