I have a question regarding a passage in Chapter X of "Calcul Stochastique et Problèmes de Martingales"J.Jacod(1979). In (10.13) they define an adapted process $X$ to the time change $\tau(t)$ as a process that is constant on the interval $[\tau_{t-},\tau_t]$ (see also Lemma 10.14).
I don't really understand what this means(my French is not that good so it might be possible that I don't understand what that definition means). I can see how this holds true for a time change driven by a continuous process, however if the time change has jumps, what would that condition imply?
Thank you .
Edit
For $\tau_t=\inf\{s: C_s<t\}$, where $C_t$ is an increasing process.
Def. 10.13
On dit qu'un processus $X$ est adapté au changement de temps $\tau$ si $X$ est constant sur chaque intervalle $[\tau_{t-},\tau_t]$.
Lemme 10.14
Il y a équivalence entre :
i) X est adapté a $\tau$;
ii) $X_{u\wedge\tau(\infty)}=X_{\tau(C_u)}$ pour tout $u\in \mathbb{R}_{+}$;
iii) $X$ est constant sur chaque intervalle $[u,v]$ tel que $C_u=C_{v-}<\infty$.
Edit for translation:
Definition 10.13 :
A process $X$ is said to be adapted to a time change $\tau$ if $X$ is constant over each interval of the form $[\tau_{t-},\tau_t]$.
Lemma 10.14 :
The following three assertions are equivalent :
i) X is adapted to $\tau$;
ii) $X_{u\wedge\tau(\infty)}=X_{\tau(C_u)}$, $\forall u\in \mathbb{R}_{+}$;
iii)$X$ is constant on each interval $[u,v]$ such that $C_u=C_{v-}<\infty$.