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

Question

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?

Answer 1

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 $$ Y_t = \sum_{k=0}^\infty 1_{\{\tau_k \le t\}} \max(0,\Delta X_{\tau_k \wedge t}). $$