{"paper":{"title":"Properties of $G$-martingales with finite variation and the application to $G$-Sobolev spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yongsheng Song","submitted_at":"2016-07-03T08:54:47Z","abstract_excerpt":"As is known, a process of form $\\int_0^t\\eta_sd\\langle B\\rangle_s-\\int_0^t2G(\\eta_s)ds$, $\\eta\\in M^1_G(0,T)$, is a non-increasing $G$-martingale. In this paper, we shall show that a non-increasing $G$-martingale could not be form of $\\int_0^t\\eta_sds$ or $\\int_0^t\\gamma_sd\\langle B\\rangle_s$, $\\eta, \\gamma \\in M^1_G(0,T)$, which implies that the decomposition for generalized $G$-It\\^o processes is unique: For $\\zeta\\in H^1_G(0,T)$, $\\eta\\in M^1_G(0,T)$ and non-increasing $G$-martingales $K, L$, if \\[\\int_0^t\\zeta_s dB_s+\\int_0^t\\eta_sds+K_t=L_t,\\ t\\in[0,T],\\] then we have $\\eta\\equiv0$, $\\zet"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.00616","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"}