{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(W_t)_{t \\ge 0}$ be a [[Wiener process|552AWX3QXN]], let $a \u003e 0$, and let $\\tau_a = \\inf\\set{t \\ge 0 : W_t = a}$ be the first hitting time of $a$, a [[stopping time|WND87FBAKC]].\n\n1. The process $\\widetilde{W}$ defined by $\\widetilde{W}_t = W_t$ for $t \\le \\tau_a$ and $\\widetilde{W}_t = 2a - W_t$ for $t \u003e \\tau_a$ is again a Wiener process.\n2. For every $t \u003e 0$,\n$$P\\Bigl(\\sup_{0 \\le s \\le t} W_s \\ge a\\Bigr) = 2\\, P(W_t \\ge a).$$\n\n## Remarks\n\nPart 1 follows from the strong Markov property: after $\\tau_a$, the process $W_{\\tau_a + s} - a$ is a Wiener process independent of the past, and so is its negative.\nFor part 2, the event $\\{\\sup_{s \\le t} W_s \\ge a\\}$ equals $\\{\\tau_a \\le t\\}$, and it splits into $\\{W_t \\ge a\\}$ and $\\{\\tau_a \\le t, W_t \u003c a\\}$.\nBy part 1 the second event has the same probability as $\\{\\widetilde{W}_t \u003e a\\} = \\{W_t \u003e a\\}$, and $P(W_t = a) = 0$.\nAs a consequence, $\\sup_{s \\le t} W_s$ has the same law as $\\abs{W_t}$, and $P(\\tau_a \\le t) = 2\\bigl(1 - \\Phi(a/\\sqrt{t})\\bigr)$, where $\\Phi$ is the standard normal distribution function.\nDifferentiating gives the density $\\frac{a}{\\sqrt{2\\pi t^3}} e^{-a^2/(2t)}$ of $\\tau_a$, which shows $\\tau_a \u003c \\infty$ almost surely but $E[\\tau_a] = \\infty$.\n\n## Explanations\n\n### L1 human\n\nOnce a Brownian path touches the level $a$, it is equally likely to end above or below $a$.\nReflecting the rest of the path in the line $y = a$ turns one kind of path into the other.\nSo the chance that the path ever reaches $a$ by time $t$ is twice the chance that it ends above $a$ at time $t$.\n\n### L2 human\n\nThe principle gives the joint law of $W_t$ and its running maximum, and the law of first passage times.\nThese are used to price barrier options and to study boundary crossing in sequential statistics.\nThe same reflection idea for simple random walks gives the ballot theorem and counts of lattice paths.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "22735FGJV6"
    ],
    "kind": "statement",
    "msc": [
      "60J65"
    ],
    "name": "Reflection principle",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7307248",
      "wikipedia": "Reflection principle (Wiener process)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Reflection_principle_(Wiener_process)"
      }
    ],
    "tag": "K6R7GHB3VS",
    "uses": [
      "552AWX3QXN",
      "WND87FBAKC",
      "05AQE0QR0D"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/K6R7GHB3VS/",
  "links": [
    {
      "field": "about",
      "target": "22735FGJV6"
    },
    {
      "field": "uses",
      "target": "552AWX3QXN"
    },
    {
      "field": "uses",
      "target": "WND87FBAKC"
    },
    {
      "field": "uses",
      "target": "05AQE0QR0D"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/K6R7GHB3VS.md",
  "msc": [
    "60J65"
  ],
  "name": "Reflection principle",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/K6R/K6R7GHB3VS.md",
  "status": null,
  "tag": "K6R7GHB3VS",
  "uri": "m2://K6R7GHB3VS",
  "url": "https://math2.tamnd.com/wiki/reflection-principle/",
  "writing": {
    "tag": "K6R7GHB3VS",
    "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
      }
    ]
  }
}