**Set-up and assumptions.** Let $(\mathscr{F}_t, t \geq 0)$ be a right-continuous complete filtration. Let $(X_t, t\geq 0 )$ be a pure jump $\mathbb{R}$-valued process with unit jumps, that is,
$$
X_t = \sum\limits _{i = 1} ^\infty I\{ \tau _i <t \},
$$
where $\{\tau _i \}$ is an a.s. increasing sequence of $(\mathscr{F}_t)$-stopping times, a.s. $\lim_{i \to \infty} \tau _i = \infty$. Assume also that $E X _t < \infty$, and that all $\tau _i$ are totally inaccessible.

We know by Doob-Meyer decomposition theorem that there exists a predictable process $(A_t)$ such that $$ X_t - A _t $$ is an $(\mathscr{F}_t)$-martingale.

Assume that we also know that for some uniformly bounded predictable continuous process $(\alpha _t, t \geq 0)$ a.s.

$$ P\big[X_{t + \Delta t} - X _t = 1 \mid \mathscr{F}_t \big] = \alpha _t \Delta t + o(\Delta t), \ \ \ \ \ \ \ (1) $$ $$ \ \ \ \ \ \ \ P\big[X_{t + \Delta t} - X _t = 0 \mid \mathscr{F}_t \big] = 1 - \alpha _t \Delta t + o(\Delta t), \ \ \ \ \ \ \ \ \ \ \ \ $$ $$ P\big[X_{t + \Delta t} - X _t > 1 \mid \mathscr{F}_t \big] = o(\Delta t). \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2) $$

**Question.** Can we prove that $A_t = \int\limits _0 ^t \alpha _s ds$?

**Thoughts.** Intuitively $\alpha _t$ should be the intensity of jumps for $(X_t)$ and $X _t - \int\limits _0 ^t \alpha _s ds$ should be a martingale. However I have not found any reference confirming this. It is proven in multiple references that the inverse implication is true, that is, if $A_t = \int\limits _0 ^t \alpha _s ds$, then (1) - (2) holds. For example, in Point Processes and Queues. Martingale Dynamics (3.5) in Chapter 2 is very similar to (1)-(2). Another example is Lemma 2.22 in Chapter 2 of Enlargement of Filtration with Finance in View. However I did not find the answer to the posted question and I don't see how to show it myself, and would very welcome suggestions or suitable references.