Proposition 3 (ABDL03): If a special semimartingale process $X$ is square integrable with respect to the natural filtration of a standard Brownian motion $W$, then one can write
$X_t - X_0 = \int_0^t \! \mu_u \, \mathrm{d}u + \int_0^t \! \sigma_u \, \mathrm{d}W_u$
where $\mu,\sigma$ are predictable processes.
(Note: I modified the statement of the proposition quite a bit.)
I know that if $X$ is a local martingale, this proposition holds with $\mu \equiv 0$. This is just the martingale representation theorem.
Q: What if $X$ is a finite variation process? Does this proposition hold with $\sigma\equiv 0$? If so, can the sufficient condition of square integrability be relaxed? Is there some standard set of necessary and sufficient conditions for when one can write a finite variation process as $\int_0^t \! \mu_u \, \mathrm{d}u$ with $\mu$ predictable?
UPDATE #1:
Partial Answer: So, if the paths of $X$ are continuously differentiable and bounded on compacts (almost surely?) then by the fundamental theorem of calculus we can write
$X_t - X_0 = \int_0^t \mu_u \, \mathrm{d}u $
where $\mu$ is the continuous, bounded derivative of $X$ (almost surely?) with respect to $t$.
Remaining Q: Have I done this right? Is the existence of a bounded, continuous derivative for the paths of $X$ (a finite variation process) necessary and sufficient for writing it as $\int_0^t \mu_u \, \mathrm{d}u$ with $\mu$ predictable?
UPDATE #2: We might be able to relax this to just differentiable (rather than continuously differentiable) since $X$ already has finite variation.