{"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 $X$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0818","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"}