Random ordinary differential equation

Remarks

For 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. Example: 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 $X_t' = -X_t + \eta_t$. Non-example: let $W$ be a Wiener process. The 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. A stochastic differential equation 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 $Z_t' = a(Z_t + W_t)$. Some authors allow $f$ to depend on $t$ and $\eta$ to have only measurable paths; then solutions are taken in the sense of Carathéodory.