Let $V$ be a cadlag positive supermartingale with the following decomposition:

$$V_t=V_0+\int_0^tH_sdX_s-K_t$$

where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with $K_0=0$. By Theorem 4.52 in "Limit Theorems for Stochastic Processes", one has

$$[V,X]_t=\langle V^c,X^c \rangle_t+\sum_{s\le t}\Delta V_s\Delta X_x$$

where $[V,X]$ denotes the quadratic co-variation, $V^c$ and $X^c$ denote respectively the continuous local martingale part of $V$ and $X$ and $\Delta V_t=V_t-V_{t-}$ and $\Delta X_t=X_t-X_{t-}$.

Could some help me calculate explicitly $[V,X]_t$, $\langle V^c,X^c \rangle_t$ and $\Delta V_s\Delta X_s$ only by $H$ and $X$? (not involving $K$!) Thanks a lot!