{"paper":{"title":"On the Quasi-Stationary Distribution of the Shiryaev-Roberts Diffusion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR","stat.ME","stat.TH"],"primary_cat":"math.ST","authors_text":"Aleksey S. Polunchenko","submitted_at":"2016-06-21T17:00:13Z","abstract_excerpt":"We consider the diffusion $(R_t^r)_{t\\ge0}$ generated by the equation $dR_t^r=dt+\\mu R_t^r dB_t$ with $R_0^r\\triangleq r\\ge0$ fixed, and where $\\mu\\neq0$ is given, and $(B_t)_{t\\ge0}$ is standard Brownian motion. We assume that $(R_t^r)_{t\\ge0}$ is stopped at $\\mathcal{S}_A^r\\triangleq\\inf\\{t\\ge0\\colon R_t^r=A\\}$ with $A>0$ preset, and obtain a closed-from formula for the quasi-stationary distribution of $(R_t^r)_{t\\ge0}$, i.e., the limit $Q_A(x)\\triangleq\\lim_{t\\to+\\infty}\\Pr(R_t^r\\le x|\\mathcal{S}_A^r>t)$, $x\\in[0,A]$. Further, we also prove $Q_A(x)$ to be unimodal for any $A>0$, and obtain "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.06658","kind":"arxiv","version":3},"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"}