Let $X=(X_t)_{t \geq 0}$ be a diffusion which solves the SDE \begin{align*} dX_t = \mu(X_t) dt + \sigma(X_t) dB_t, \end{align*} and I have that $X$ is a Markov process. The other process is $S_t = \max_{0 \leq r \leq t}{X_r} \lor s$ for which it is assumed that it is started at $s \geq x$ under $P^{x,s}$. Now I want to show that $Y_t=(X_t, S_t)$ is a two-dimensional Markov process.
I tried to imitate the answer given here but it didn't work.
