{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_{t \\ge 0}, P)$ be a probability space with a [[filtration|130FCDWRT1]], and let $W$ be a one-dimensional [[Wiener process|22735FGJV6]] with respect to this filtration.\nLet $\\tau \u003e 0$ and $\\Delta = \\set{(t, s) : 0 \\le s \\le t \\le \\tau}$.\nLet $a, b \\colon \\Delta \\times \\R \\to \\R$ be measurable functions, and let $X_0$ be a continuous [[adapted process|YRNFE016FF]] on $[0, \\tau]$.\nThe *stochastic Volterra equation* with coefficients $a$, $b$ and free term $X_0$ is\n$$X(t) = X_0(t) + \\int_0^t a(t, s, X(s)) \\, ds + \\int_0^t b(t, s, X(s)) \\, dW(s), \\qquad 0 \\le t \\le \\tau,$$\nwhere the last integral is an Itô integral, see [[Itô calculus|JA95NH206E]].\nA *solution* is a continuous adapted process $X$ on $[0, \\tau]$ such that, for every $t \\in [0, \\tau]$:\n\n1. almost surely $\\int_0^t \\abs{a(t, s, X(s))} \\, ds \u003c \\infty$ and $\\int_0^t b(t, s, X(s))^2 \\, ds \u003c \\infty$;\n2. the equation holds almost surely.\n\n## Remarks\n\nIf $a(t, s, x) = \\tilde a(s, x)$ and $b(t, s, x) = \\tilde b(s, x)$ do not depend on $t$ and $X_0(t) = \\xi$ is constant in time, the equation is the integral form of the [[stochastic differential equation|B219G9SHJA]] $dX = \\tilde a(t, X) \\, dt + \\tilde b(t, X) \\, dW$, $X(0) = \\xi$.\nIf $b = 0$, the equation is a deterministic [[Volterra integral equation|DND8CBEVXT]] for each path.\nSystems with values in $\\R^n$ and several Wiener processes are defined in the same way.\n\nExample: $a = 0$, $b(t, s, x) = x$ and $X_0 = 1$.\nThe equation is $X(t) = 1 + \\int_0^t X(s) \\, dW(s)$, and $X(t) = \\exp(W(t) - t/2)$ is a solution.\nIndeed, Itô's formula applied to $f(t, w) = e^{w - t/2}$ gives $dX = X \\, dW + \\big(-\\tfrac{1}{2} + \\tfrac{1}{2}\\big) X \\, dt = X \\, dW$.\n\nExample with memory: $a = 0$, $X_0 = 0$ and $b(t, s, x) = (t - s)^{H - 1/2}$ with $H \u003e 0$.\nClause 1 holds since $\\int_0^t (t - s)^{2H - 1} \\, ds = t^{2H} / (2H) \u003c \\infty$, and the solution is the process $X(t) = \\int_0^t (t - s)^{H - 1/2} \\, dW(s)$, up to a choice of a continuous version.\n\nNon-example: for $b(t, s, x) = (t - s)^{-1/2}$, the case $H = 0$, the integral $\\int_0^t (t - s)^{-1} \\, ds$ diverges, so clause 1 fails for every $t \u003e 0$ and the equation has no solution.\n\n## Explanations\n\n### L1 human\n\nIt is a Volterra equation in which part of the input arrives through random kicks $dW$, whose effect may depend on the time elapsed since they happened.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "KA95J9V6AS",
    "kind": "definition",
    "msc": [
      "60H20",
      "45R05",
      "45D05"
    ],
    "name": "Stochastic Volterra equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Stochastic_differential_equation"
      }
    ],
    "tag": "5C3411G4C0",
    "uses": [
      "DND8CBEVXT",
      "22735FGJV6",
      "JA95NH206E",
      "YRNFE016FF",
      "130FCDWRT1"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/5C3411G4C0/",
  "links": [
    {
      "field": "defines",
      "target": "KA95J9V6AS"
    },
    {
      "field": "uses",
      "target": "DND8CBEVXT"
    },
    {
      "field": "uses",
      "target": "22735FGJV6"
    },
    {
      "field": "uses",
      "target": "JA95NH206E"
    },
    {
      "field": "uses",
      "target": "YRNFE016FF"
    },
    {
      "field": "uses",
      "target": "130FCDWRT1"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/5C3411G4C0.md",
  "msc": [
    "60H20",
    "45R05",
    "45D05"
  ],
  "name": "Stochastic Volterra equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5C3/5C3411G4C0.md",
  "status": null,
  "tag": "5C3411G4C0",
  "uri": "m2://5C3411G4C0",
  "url": "https://tamnd.github.io/math2/wiki/stochastic-volterra-equation-definition/",
  "writing": {
    "tag": "5C3411G4C0",
    "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
      }
    ]
  }
}