I am self-studying "Introduction to Stochastic Integration" by Hui-Hsiung Kuo and have some doubts on the lemma. In the book it first defines that the Markov property on page 198 as follow
Definition 10.5.4. A stochastic process $X_t$, $a \le t \le b$, is said to satisfy the Markov property if for any $a \le t_1 < t_2 < ··· < t_n < t \le b$, the equality $$P(X_t \le x | X_{t_1}, X_{t_2} ,\dots, X_{t_n}) = P(X_t \le x | X_{t_n})$$
Then there is a lemma
Lemma 10.5.9. Suppose a stochastic process $X_t$, $a \le t \le b$, is adapted to a filtration $\{\mathcal{F}_t; a \le t \le b\}$ and satisfies the condition $$P(X_t \le x | \mathcal{F}_s) = P(X_t \le x | X_s),\quad \forall s<t, x\in\mathbb{R}$$ Then $X_t$ is a Markov process.
It seems that in the lemma, we just need $X_t$ adapted to the filtration $\mathcal{F}_t$($\mathcal{F}_t$ does not necessarily need to be generated by $X_t$). Then my question is that: Conversely, if we are given $X_t$ is a Markov process and want to show the condition, i.e., $P(X_t \le x | \mathcal{F}_s) = P(X_t \le x | X_s)$, do we need a restriction on $\mathcal{F}_t$ that $\mathcal{F}_t$ needs to be generated by $X_t$?