{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $p \u003e 0$.\nThere are constants $0 \u003c c_p \\le C_p \u003c \\infty$, depending only on $p$, with the following property.\nLet $(M_t)_{t \\ge 0}$ be a continuous [[local martingale|ZWJ9G89W3T]] with $M_0 = 0$, with [[quadratic variation|ZBYC3MFW89]] $[M]$, and let $\\tau$ be a [[stopping time|WND87FBAKC]].\nLet $M^*_\\tau = \\sup_{0 \\le s \\le \\tau} \\abs{M_s}$.\nThen\n$$c_p\\, E\\bigl[[M]_\\tau^{p/2}\\bigr] \\le E\\bigl[(M^*_\\tau)^p\\bigr] \\le C_p\\, E\\bigl[[M]_\\tau^{p/2}\\bigr].$$\n\n## Remarks\n\nFor $p = 2$ the inequality follows from [[Doob's martingale inequality|4G53E7BRXF]] in its $L^2$ form and the identity $E[M_\\tau^2] = E[[M]_\\tau]$ for suitable $\\tau$, with $c_2 = 1$ and $C_2 = 4$.\nFor other $p$, one standard proof applies Itô's formula to $\\abs{M}^p$ or uses a good-$\\lambda$ inequality comparing the laws of $M^*$ and $[M]^{1/2}$.\nExample: for a Wiener process $W$ and a fixed time $t$, $[W]_t = t$, so $E[\\sup_{s \\le t} \\abs{W_s}^p]$ is between $c_p t^{p/2}$ and $C_p t^{p/2}$.\nFor càdlàg local martingales the inequality holds for $p \\ge 1$, with the same form.\nThe constants do not depend on $M$ or $\\tau$, and this uniformity is what makes the inequality useful.\n\n## Explanations\n\n### L1 human\n\nThe size of the largest swing of a fair game is comparable to the square root of its accumulated squared fluctuations.\nBoth quantities measure how much the game has moved, and the inequality says they agree up to constant factors.\nIt holds in every $L^p$ norm.\n\n### L2 human\n\nThe inequality lets one bound moments of a stochastic integral $\\int_0^t H_s \\, dW_s$ by moments of $\\int_0^t H_s^2 \\, ds$.\nIt is a key estimate in proving existence, uniqueness and moment bounds for solutions of stochastic differential equations.\nIt also shows that a continuous local martingale with $E[[M]_\\infty^{1/2}] \u003c \\infty$ is a uniformly integrable martingale.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "XC2YPC2BSW",
      "X4DHMZD5JA"
    ],
    "kind": "statement",
    "msc": [
      "60G44",
      "60E15"
    ],
    "name": "Burkholder–Davis–Gundy inequality",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quadratic_variation"
      }
    ],
    "tag": "SWHVQD9YEB",
    "uses": [
      "ZWJ9G89W3T",
      "ZBYC3MFW89",
      "WND87FBAKC",
      "0DCT0QZVQ1"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/SWHVQD9YEB/",
  "links": [
    {
      "field": "about",
      "target": "XC2YPC2BSW"
    },
    {
      "field": "about",
      "target": "X4DHMZD5JA"
    },
    {
      "field": "uses",
      "target": "ZWJ9G89W3T"
    },
    {
      "field": "uses",
      "target": "ZBYC3MFW89"
    },
    {
      "field": "uses",
      "target": "WND87FBAKC"
    },
    {
      "field": "uses",
      "target": "0DCT0QZVQ1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/SWHVQD9YEB.md",
  "msc": [
    "60G44",
    "60E15"
  ],
  "name": "Burkholder–Davis–Gundy inequality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SWH/SWHVQD9YEB.md",
  "status": null,
  "tag": "SWHVQD9YEB",
  "uri": "m2://SWHVQD9YEB",
  "url": "https://math2.tamnd.com/wiki/burkholder-davis-gundy-inequality/",
  "writing": {
    "tag": "SWHVQD9YEB",
    "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
      }
    ]
  }
}