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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]] and let $[a, b]$ be a compact interval.\nLet $k \\colon [a, b] \\times [a, b] \\times \\Omega \\to \\R$ and $h \\colon [a, b] \\times \\Omega \\to \\R$ be [[measurable functions|YYNCJSR0PJ]].\nHere the domains carry the product of the Borel $\\sigma$-algebras and $\\mathcal{F}$.\nThe *random (linear Fredholm) integral equation* with random kernel $k$ and random free term $h$ is\n$$x(t, \\omega) = h(t, \\omega) + \\int_a^b k(t, s, \\omega) \\, x(s, \\omega) \\, ds, \\qquad t \\in [a, b], \\ \\omega \\in \\Omega.$$\nA *random solution* is a measurable function $x \\colon [a, b] \\times \\Omega \\to \\R$ with the following property.\nFor $P$-almost every $\\omega$, the integral converges absolutely and the equation holds, for every $t \\in [a, b]$.\n\n## Remarks\n\nRandom Volterra equations, with $\\int_a^t$ in place of $\\int_a^b$, and nonlinear random equations such as $x(t, \\omega) = h(t, \\omega) + \\int_a^b k(t, s, \\omega) f(s, x(s, \\omega)) \\, ds$ are defined in the same way.\nA random solution is a [[stochastic process|0QH5ENKZM1]] indexed by $t \\in [a, b]$.\nThe integral is an ordinary Lebesgue integral, computed separately for each $\\omega$.\n\nExample: $[a, b] = [0, 1]$, $Y$ a random variable with values in $[0, 1]$, $k(t, s, \\omega) = Y(\\omega)/2$ and $h = 1$.\nA constant $x(t, \\omega) = c(\\omega)$ solves the equation when $c = 1 + Y c / 2$, that is, $c = 2/(2 - Y)$.\nThis $c$ is measurable, so $x(t, \\omega) = 2/(2 - Y(\\omega))$ is a random solution.\n\nNon-example: the equation $x(t) = 1 + \\int_0^t x(s) \\, dW(s)$, with $W$ a [[Wiener process|22735FGJV6]], is not a random integral equation in this sense.\nAlmost every path of $W$ has unbounded variation, so the integral cannot be computed path by path as a Lebesgue–Stieltjes integral.\nIt is an Itô integral, and the equation is a [[stochastic Volterra equation|KA95J9V6AS]].\n\n## Explanations\n\n### L1 human\n\nIt is a family of integral equations, one for each outcome $\\omega$ of a random experiment, solved outcome by outcome.\n",
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