Joint filtration vs join of filtrations for counting processes

Question

Let $X,Y$ be counting processes without common jumps, i.e. $\Delta X \Delta Y = 0$ $P$-a.s. Denote by $\mathcal{F}_t^X$ and $\mathcal{F}^Y_t$ the filtrations generated by $X$ and $Y$, respectively. Is it true that $\sigma(X_s,Y_s:s\leq t)=\mathcal{F}_t^X \vee \mathcal{F}_t^Y$? I.e. is the $\sigma$-algebra generated by the two processes jointly up to $t$ equal to the join of the two $\sigma$-algebras generated by the respective processes up to $t$?