{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] $(\\mathcal{F}_t)_{t \\ge 0}$ that is right-continuous and complete.\nLet $(X_t)_{t \\ge 0}$ be a [[submartingale|EQ055QMACW]] with càdlàg paths, that is, right-continuous paths with left limits.\nAssume $X$ is of class (D): the family of random variables $X_\\tau$, where $\\tau$ runs over the finite [[stopping times|WND87FBAKC]], is uniformly integrable.\nThen there is a unique decomposition $X_t = M_t + A_t$ for $t \\ge 0$ with these properties:\n\n1. $(M_t)$ is a uniformly integrable càdlàg [[martingale|0CDVK9RQBH]].\n2. $(A_t)$ is a predictable, non-decreasing càdlàg process with $A_0 = 0$.\n3. $E[A_\\infty] \u003c \\infty$, where $A_\\infty = \\lim_{t \\to \\infty} A_t$.\n\nUniqueness is up to indistinguishability.\n\n## Remarks\n\nHere a process is predictable if it is measurable for the σ-algebra on $[0, \\infty) \\times \\Omega$ generated by the left-continuous [[adapted|YRNFE016FF]] processes.\nOne proof applies the discrete [[Doob decomposition theorem|YHSSQ67GDS]] on dyadic time grids and shows that the predictable parts converge, using class (D) for uniform integrability.\nExample: for a Wiener process $W$ and $T \u003e 0$, the stopped process $X_t = W_{t \\wedge T}^2$ is a submartingale of class (D), and its decomposition is $M_t = W_{t \\wedge T}^2 - t \\wedge T$, $A_t = t \\wedge T$.\nWithout class (D) a decomposition still exists locally, with $M$ a [[local martingale|ZWJ9G89W3T]]; this is used for $W_t^2 = (W_t^2 - t) + t$ on $[0, \\infty)$.\nPredictability is what makes $A$ unique.\nFor a Poisson process $N$ with rate $1$, both $N_t = 0 + N_t$ and $N_t = (N_t - t) + t$ split $N$ into a martingale and an increasing process.\nOnly $A_t = t$ is predictable.\n\n## Explanations\n\n### L1 human\n\nA process that tends to rise splits into a fair game plus a rising trend.\nThe trend can be foreseen an instant before it happens, and the fair game holds all the surprise.\nThis is the continuous-time version of the Doob decomposition for sequences.\n\n### L2 human\n\nThe theorem is the source of the predictable quadratic variation $\\langle M \\rangle$ of a square-integrable martingale $M$, the compensator $A$ of $M^2$.\nIt underlies the construction of the stochastic integral with respect to martingales and the theory of [[semimartingales|MZNFPJJMY9]].\nCompensators of counting processes, such as $t$ for the Poisson process, come from the same theorem and are central in point process theory.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "6M852B4RA8",
      "0D0GBNDKDH"
    ],
    "kind": "statement",
    "msc": [
      "60G44",
      "60G48"
    ],
    "name": "Doob–Meyer decomposition theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q5296972",
      "wikipedia": "Doob–Meyer decomposition theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Doob%E2%80%93Meyer_decomposition_theorem"
      }
    ],
    "tag": "SX7ZPP72EM",
    "uses": [
      "EQ055QMACW",
      "0CDVK9RQBH",
      "HNK1Y62G7V",
      "WND87FBAKC",
      "YRNFE016FF"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/SX7ZPP72EM/",
  "links": [
    {
      "field": "about",
      "target": "6M852B4RA8"
    },
    {
      "field": "about",
      "target": "0D0GBNDKDH"
    },
    {
      "field": "uses",
      "target": "EQ055QMACW"
    },
    {
      "field": "uses",
      "target": "0CDVK9RQBH"
    },
    {
      "field": "uses",
      "target": "HNK1Y62G7V"
    },
    {
      "field": "uses",
      "target": "WND87FBAKC"
    },
    {
      "field": "uses",
      "target": "YRNFE016FF"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/SX7ZPP72EM.md",
  "msc": [
    "60G44",
    "60G48"
  ],
  "name": "Doob–Meyer decomposition theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SX7/SX7ZPP72EM.md",
  "status": null,
  "tag": "SX7ZPP72EM",
  "uri": "m2://SX7ZPP72EM",
  "url": "https://math2.tamnd.com/wiki/doob-meyer-decomposition-theorem/",
  "writing": {
    "tag": "SX7ZPP72EM",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}