Doob-Meyer decomposition of number of times the diffusion cross a barrier

Question

Let $X_t=\mu t+\sigma B_t$ for all $t\geq 0$ be a drifted Brownian motion, where $B$ is a standard Brownian motion, $\mu\in\mathbb{R}$ and $\sigma>0$. Let $c\in\mathbb{R}$ and $L_t=\sum_{0<s\leq t}\max(0,\Delta\mathbb{1}_{\{X_s \geq c\}})$ be the number of times that $X$ crosses $c$ from below. As $L$ is a non-decreasing process, hence, a sub-martingale. Then by Doob-Meyer's theorem there should be a unique decomposition of $L$ such that $$ L_t=M_t+A_t $$ a.s., where $M_t$ is a martingale and $A_t$ is a non-decreasing process.

The question is, how to identify $M_t$ and $A_t$?

Answer 1

Since $L$ itself is a non-decreasing process, $A_t = L_t$ and $M_t = 0$ is such a decomposition. By uniqueness of the Doob-Meyer decomposition, this is the only such decomposition.