{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-martingale",
  "semantic_family_id": "sf-martingale",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限Martingale"
  },
  "summary": {
    "zh-Hant": "相對同一filtration適應且可積的有限過程，若每一步下一值條件期望等於現在值a.s.，則為Martingale。"
  },
  "statement": {
    "zh-Hant": "在共同有限機率空間上完整核filtration nesting、adaptedness、可積性及每個t的E[X_{t+1}|F_t]=X_t。零質量資訊block只a.s.決定；不推出獨立增量、Markov、optional stopping或收斂。"
  },
  "formula": {
    "tex": "M(X,F)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>M</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "M",
        "glyph": "M"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        },
        {
          "type": "symbol",
          "name": "F",
          "glyph": "F"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "b42eea83a030787af996481d3725dfc677e1ab92761aa2ee957281ad36a6b11d"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "精確有理質量、partition filtration與process值綁同一有限樣本空間。"
    },
    {
      "id": "a2",
      "zh-Hant": "主要模型的conditioning blocks為正質量；零質量只記a.s.邊界。"
    }
  ],
  "symbols": [
    {
      "token": "M",
      "role": "finite_martingale",
      "role_zh": "有限Martingale",
      "scope": "representative_formula"
    },
    {
      "token": "F",
      "role": "finite_filtration",
      "role_zh": "同一filtration",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-filtration",
      "title_zh": "有限filtration",
      "reason_zh": "資訊序列須完整嵌套。"
    },
    {
      "id": "mko-conditional-expectation",
      "title_zh": "有限條件期望",
      "reason_zh": "逐步martingale等式。"
    },
    {
      "id": "mko-discrete-stochastic-process",
      "title_zh": "離散隨機過程",
      "reason_zh": "X_t按時間索引。"
    },
    {
      "id": "mko-expectation",
      "title_zh": "期望值",
      "reason_zh": "可積與條件平均。"
    },
    {
      "id": "mko-measurable-function",
      "title_zh": "可測函數",
      "reason_zh": "adaptedness逐時核對。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先核filtration與adaptedness。",
      "再對每個time-block算下一值條件平均。",
      "期望不變是後果，不是完整martingale定義。"
    ],
    "common_misconception_zh": "只看總期望不變就稱martingale，或自動推出independent increments與optional stopping。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理指定有限時間模型；不宣稱收斂或停止定理。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入submartingale、stopping time與optional stopping邊界。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-martingale.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-10-30 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}