Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\geq 0}, P)$ is a Probability space, and $\{X_t\}_{t\geq 0}$ is a $1$-dimentional continuous Markov process.
I want to know whether $Y_t := \text{max}_{0\leq u \leq t}X_u$ is a Markov process or not.
I predict $Y_t$ is a Markov process because the information needed to determine the value of $Y_t$ within $\mathcal{F}_t$ is based on when does $X_u$ attains its maximum value for $u \in [0, t]$. In this case, it is sufficient to consider the sigma-algebra generated by $\max_{0 \leq u \leq t} X_u$.
All I have to prove is that $\forall \Gamma \in \mathcal{B}(\mathbb{R})$ and $\forall A \in \mathcal{F}_s$, $$ P\left(\{\text{max}_{0\leq u \leq t + s}X_u \in \Gamma\} \cap A\right) = E\left[P\left(\text{max}_{0\leq u \leq t + s}X_u \in \Gamma \,| \,\text{max}_{0\leq u \leq s }X_u\right), A\right] $$ but what should I do after this?