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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]].\nLet $\\eta = (\\eta_t)_{t \\ge 0}$ be a [[stochastic process|0QH5ENKZM1]] with values in $\\R^m$ whose sample paths $t \\mapsto \\eta_t(\\omega)$ are continuous.\nLet $f \\colon \\R^n \\times \\R^m \\to \\R^n$ be continuous.\nThe *random ordinary differential equation* driven by $\\eta$ with right side $f$ is the family of [[ordinary differential equations|0NT64JV96A]]\n$$\\frac{dx}{dt} = f(x, \\eta_t(\\omega)), \\qquad \\omega \\in \\Omega.$$\nA *solution* is a process $(X_t)_{t \\ge 0}$ such that for $P$-almost every $\\omega$ the path $t \\mapsto X_t(\\omega)$ is continuously differentiable and solves the equation for that $\\omega$.\n\n## Remarks\n\nFor fixed $\\omega$ the right side $(t, x) \\mapsto f(x, \\eta_t(\\omega))$ is continuous, so the usual existence theory for ordinary differential equations applies path by path.\nExample: with $n = m = 1$ and $f(x, y) = -x + y$, the process $X_t = e^{-t} X_0 + \\int_0^t e^{-(t - s)} \\eta_s \\, ds$ is a solution; differentiating gives $X_t' = -X_t + \\eta_t$.\nNon-example: let $W$ be a [[Wiener process|22735FGJV6]].\nThe equation $dX_t = X_t \\, dW_t$ is not a random ordinary differential equation as written, since the noise enters through $dW_t$ and the paths of $W$ are not differentiable.\nA [[stochastic differential equation|B219G9SHJA]] with additive noise, $dX_t = a(X_t) \\, dt + dW_t$, becomes a random ordinary differential equation after the change of variables $Z_t = X_t - W_t$: then $Z_t' = a(Z_t + W_t)$.\nSome authors allow $f$ to depend on $t$ and $\\eta$ to have only measurable paths; then solutions are taken in the sense of Carathéodory.\n",
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