Let $f$ be a Lipschitz continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $W$ be a standard Brownian motion. I don't know any stochastic calculus (nothing about stochastic integrals, nothing about Itō's lemma), and I'm trying to prove that, for any fixed $\omega$, there exists a solution of the stochastic integral equation $$ \xi(T)=\xi_0+\int_0^Tf(\xi(t))dt+W(T) $$ a.s. for all $T\ge 0$.
I'm far from proving this, but it should be doable without any knowledge of stochastic calculus? Perhaps there's a useful analogue to the method of characteristics, i.e., would it be of any use to compute $ \lim_{h\to 0}h^{-1}\mathbb{E}(\xi(T+h)-\xi(T)|\xi(T)=c)? $ Any hints would be greatly appreciated!
(I've added the stochastic calculus and analysis tags, even though I know nothing about stochastic calculus, and I'm trying to prove this with more elementary methods.)
Consider the application $$ F(\xi) : F(\xi)(T)=\xi_0+\int_0^Tf(\xi(t))dt+W(T) $$
$$ \left[ F(\xi) - F(\zeta)\right] (T) = \int_0^T \left[f(\xi(t)) - f(\zeta(t))\right] dt $$ under the same assumptions as in the Cauchy Lipschitz theorem, you can prove that (for the right space and right norm) $F$ is Lipschitz with constant $<1$ (the proof is exactly the same, as the stochastic terms have cancelled); hence $F$ has an only fixed point.