Suppose on some compact time interval $[0,T]$ I have an SDE with sufficiently nice coefficients: $$ dX_t = \sigma(X_t) dB_t + \mu(X_t) dt $$ Is there any general statement that shows some form of pathwise quantitatively continuous dependence of the solution $X_t$ on $B_t$? What I mean would e.g. be a statement of the form: $$ \sup_{t \in [0,T]}\vert X_t \vert \leq C(T) \sup_{t \in [0,T]}\vert B_t \vert $$ , holding trajectory by trajectory, or some variant thereof?
I'm phrasing the question in general, but the SDE of interest comes from the unitary Brownian motion on the unitary group.
$$X_t = \mu t+\sigma W_t \iff |X_t-\mu t|=\sigma |B_t|$$ Because $$|X_t| = |(X_t-\mu t)+\mu t| \le |X_t-\mu t|+ |\mu| t$$ We deduce $$|X_t| \le |\mu| t + \sigma |B_t|$$ $$\Longrightarrow \color{red}{ \sup_{t \in [0,T]}\vert X_t \vert \le |\mu| T+ \sigma \cdot \sup_{t \in [0,T]}\vert B_t \vert} \tag{1}$$ The inequality $(1)$ is tight as the equality can occur. Indeed, if the trajectory $(B_t(\omega))_{t\in[0,T]}$ correspond to the event $\omega$ satisfy $B_t(\omega)=0 \hspace{0.5cm} \forall t\in[0,T]$, then the equality of $(1)$ holds.