Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition:
$$V_t=V_0+\int_0^tH_sdX_s-K_t$$
where $H$ is a $X$-integrable predictable process and $K$ is an adapted increasing process with $K_0=0$. How can we characterize $H$ by our observations $V$ and $X$, i.e. how to express $H$ by $V$ and $X$? Thanks a lot for the reply!