Suppose $X$ is progressively measurable with respect to the filtration of a Brownian motion $B$ and $L^2$. Suppose that $[X,B](T)=C$ for some real constant $C$, then is it true that $X$ is linear in $B$?
Because $X$ is $L^2$ we can use the martingale representation theorem to get that $$X(t)=f(t)+\int_0^t\phi(s) dB(s),$$
for some progressively measurable $\phi$.
Therefore the quadratic covariation is
$$\int_0^T \phi(s) ds=C.$$
If this is constant (independent of $B$), then is it true that $\phi(s)$ is independent of $B$? If not, what can we assume on $X$?