I am looking for a proof of the existence and uniquenes of ODE's of the type:
\begin{equation} \dot{f}(t,x,y) = F(h(t,x), f(t,x,y)), \end{equation}
where $f : T \times X \times Y \rightarrow Y, h : T \times X \rightarrow X$ and $F : X \times Y \rightarrow Y$, i.e. a pathwise random differential equation:
\begin{equation} \dot{f}_t = F(h_t (x), f_t). \end{equation} My motivation is the study of random dynamical systems (RDS), which is generated by equations of the type
\begin{equation} \dot{\phi}(t, \omega, x) = f(\theta (t, \omega), \phi (t, \omega, x)), \end{equation}
where $\phi : T \times \Omega \times X \rightarrow X$ and $\theta : T \times \Omega \rightarrow \Omega$, for $T$ some semigroup, $X$ a (measurable/topological) space and $\Omega$ a probability space.
These are treated in the seminal book on RDS by L. Arnold (chapter 2), but he refers to a rather obscure, German 1983 paper for the existence & uniqueness part.
Any help is greatly appreciated.