# Is the sum of positive jumps from a jump process adapted again?

Let $$(\Omega,\mathcal F,\{\mathcal F_t\}_{t\in [0;T]},\mathbb P)$$ be a filtered probability space satisfying the usual conditions and let $$\{X_t\}_{t\in[0;T]}$$ be an adapted stochastic process, whose paths are right-continuous step functions with finitely many steps. As usual, $$\Delta X_t := X_t - X_{t-}$$ and write $$X_{0-} := 0$$. Define $$Y_t := \sum_{s \in [0;t], \Delta X_s > 0} \Delta X_s.$$

Is $$Y$$ necessarily adapted and how can I prove that?

Finally, I think that I proved that $$Y$$ is adapted. We define $$\tau_k$$ as the time of the $$k$$-th jump ($$\tau_0 := 0$$). This is equivalent to the recursive definition $$\tau_{k+1} := \inf \{ t \in (\tau_k;T] | X_t \ne X_{\tau_k} \}.$$ To prove that $$\tau_{k+1}$$ is a stopping time, note that $$\{ \tau_{k+1} > t \} = \{ \tau_k \ge t \} \cup ( \{ \tau_k < t \} \cap \{ \forall s\in(\tau_k;t] \colon X_s = X_{\tau_k} \} )$$ and $$\{ \forall s\in(\tau_k;t] \colon j_s = j_{\tau_k} \} = \bigcap_{s \in (\mathbb Q \cap [0;t)) \cup \{t\}} ( \{ \tau_k \ge s \} \cup \{ X_s = X_{\tau_k \wedge t} \} ) \in \mathcal F_t.$$
Now $$Y_t = \sum_{k=0}^\infty 1_{\{\tau_k \le t\}} \max(0,\Delta X_{\tau_k \wedge t}).$$