There is a martingale representation theorem
If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that $ M_t = M_0 + \int_0^t \sigma(B_s)dB_s, \text{ a.s., for all } t \ge 0. $
Is there an equivalent for continuous process of finite variation?
Something like
If $V$ is a continuous process of finite variation, there is a Brownian motion $B$ and a cadlag adapted function $\mu$ such that $ V_t = V_0 + \int_0^t \mu(B_s)ds, \text{ a.s., for all } t \ge 0. $