{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_{t \\in T}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]], where $T \\subseteq [0, \\infty)$.\nLet $X = (X_t)_{t \\in T}$ be a [[stochastic process|0QH5ENKZM1]] with values in a measurable space $S$, and let $\\tau$ be a [[stopping time|WND87FBAKC]].\nThe *stopped process* $X^\\tau = (X^\\tau_t)_{t \\in T}$ is defined by\n$$X^\\tau_t(\\omega) = X_{\\min(t, \\tau(\\omega))}(\\omega) \\quad \\text{for } t \\in T \\text{ and } \\omega \\in \\Omega.$$\n\n## Remarks\n\nThe notation $t \\wedge \\tau = \\min(t, \\tau)$ is standard, so $X^\\tau_t = X_{t \\wedge \\tau}$.\nOn the event $\\set{\\tau = \\infty}$, the stopped process equals $X$.\n\nIn discrete time, if $X$ is [[adapted|YRNFE016FF]], then so is $X^\\tau$, because\n$$X^\\tau_n = X_0 + \\sum_{k=1}^{n} \\mathbf{1}_{\\set{\\tau \\ge k}} (X_k - X_{k-1})$$\nfor real-valued $X$, and $\\set{\\tau \\ge k} = \\Omega \\setminus \\set{\\tau \\le k - 1} \\in \\mathcal{F}_{k-1}$.\nThe same formula shows that a stopped [[martingale|0D0GBNDKDH]] is a martingale.\nIn continuous time, the stopped process of a progressively measurable process is progressively measurable.\n\nExample: let $S$ be a simple symmetric [[random walk|04VZE3FZXF]] started at $0$ and let $\\tau$ be the first time with $\\abs{S_n} = 5$.\nThen $\\abs{S^\\tau_n} \\le 5$ for all $n$, so $S^\\tau$ is a bounded martingale.\n\nNon-example: if $\\tau$ is not a stopping time, the formula still defines a process, but it need not be adapted.\nTake the walk above and let $\\sigma$ be the last visit to $0$ before time $10$.\nThe value $S^\\sigma_1$ equals $S_1$ if the walk returns to $0$ between times $1$ and $10$, and $0$ otherwise.\nThis is not determined by $\\mathcal{F}_1$, so $S^\\sigma$ is not adapted.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "NG35MGYDZR",
    "kind": "definition",
    "msc": [
      "60G40",
      "60G05"
    ],
    "name": "Stopped process",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Stopped_process"
      }
    ],
    "tag": "NTFND9FRDN",
    "uses": [
      "0QH5ENKZM1",
      "HNK1Y62G7V",
      "WND87FBAKC",
      "YRNFE016FF"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/NTFND9FRDN/",
  "links": [
    {
      "field": "defines",
      "target": "NG35MGYDZR"
    },
    {
      "field": "uses",
      "target": "0QH5ENKZM1"
    },
    {
      "field": "uses",
      "target": "HNK1Y62G7V"
    },
    {
      "field": "uses",
      "target": "WND87FBAKC"
    },
    {
      "field": "uses",
      "target": "YRNFE016FF"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/NTFND9FRDN.md",
  "msc": [
    "60G40",
    "60G05"
  ],
  "name": "Stopped process",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NTF/NTFND9FRDN.md",
  "status": null,
  "tag": "NTFND9FRDN",
  "uri": "m2://NTFND9FRDN",
  "url": "https://math2.tamnd.com/wiki/stopped-process-definition/",
  "writing": {
    "tag": "NTFND9FRDN",
    "kind": "definition",
    "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
      }
    ]
  }
}