{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-finite-filtration"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-finite-filtration-01","title":{"zh-Hant":"有限 filtration：以分割細化核對資訊嵌套，並區分 adapted 與獨立","en":"Finite filtration: checking information nesting by partition refinement, and separating it from adaptedness and independence"},"summary":{"zh-Hant":"在 Ω={0,1,2,3}（兩次擲幣：0=正正、1=正反、2=反正、3=反反）上，以分割編碼 P_0、P_1、P_2 表示時刻 0、1、2 的 σ 代數。證明「較晚分割細化較早分割 ⇔ σ 代數包含」，逐相鄰時刻完整核對，得出 (F_t) 是 filtration；給出不是 filtration 的分割序列；並證明 filtration 既不使每個過程 adapted，也不推出獨立。","en":"On Ω={0,1,2,3} (two coin tosses: 0=HH, 1=HT, 2=TH, 3=TT), represent the σ-algebras at times 0, 1, 2 by partition rows P_0, P_1, P_2. Prove that 'the later partition refines the earlier one ⇔ the σ-algebras are nested', check every adjacent pair of times completely and conclude that (F_t) is a filtration; give a sequence of partitions that is not a filtration; and show that a filtration neither makes every process adapted nor implies independence."},"mko_ids":["mko-finite-filtration"]}],"title":"Concept derivation guides"}