Given a filtered probability space $(\Omega, \mathbb{F}, \{\mathcal{F}_t\}_{0\le t\le 1}, \mathbb{P})$, let $X$ be a cadlag martingale and $V$ be cadlag supermartingale. Suppose $V$ has the following decomposition: there exists a $X$-integrable predictable process $H$ s.t.

$$V_t=V_0+\int_0^tH_sdX_s-C_t,~ \forall t\in [0,1]$$

where $C$ is an adapted increasing process (not necessarily predictable!) with $C_0=0$. Now if we have another predictable process $H'$ s.t.

$$V_t\le V_{t-}+H'_t(X_t-X_{t-}), \forall t\in [0,1]$$

where $V_{t-}$ and $X_{t-}$ denote the left limit. Could we say that $H=H'$ a.s. or

$$\int_0^tH_sdX_s\le \int_0^tH'_sdX_s, \forall t\in [0,1]$$

Many thanks for the reply!