Brownian noise perturbing a differential equation The following one-degree-of-freedom oscillator is given;
$$\ddot{x}+kx=w(t),$$
where, $k>0$ and $w(.)$ is a Brownian noise perturbing the system.
Assume we want to study boundedness of the solution of this differential equation without resort to stochastic analysis tools. For this, can we assume that $w(.)$ is an unknown, Lebesgue measurable and bounded function of time so that
$$|w(t)|\leq w_0,\ \forall t$$
then, we just use the standrad Lyapunov-based methods?
I just want to prove boundedness of the solution in a deterministic framework without getting involved in stochastic properties/analysis of the Brownian noise.

You have to make further assumptions on $w$, since, for instance, the solution of
$$ \ddot x + kx = \sin(t\sqrt{k}) $$
is not bounded, even if $\sin(t\sqrt{k})$ is smooth and bounded. You have to avoid resonance, i.e. completely avoid a particular frequency in the spectrum of your brownian noise.
