I have a question about Markov Process:
Let $X_t=(X_t^1, X_t^2,..., X_t^n)$ be a Markov process with regard to the filtration $\mathcal{F}_t$, let $Y_t:=\max_{1\leq k\leq n}X_t^k$, then is $Y_t$ a Markov process with respect to the filtration $\sigma(Y_s, s\leq t)$?
I believe the answer is no, but could someone give me a counterexample? Many thanks!
Consider two independent standard random walks in continuous time $(X^1_t)$ and $(X^2_t)$. Then $Y_t=\max(X^1_t,X^2_t)$ jumps from $Y_t=y$ to $y+1$ at rate $2$ if $X^1_t=X^2_t$ and to $y+1$ or to $y-1$, both at rate $1$, otherwise. The trouble is that the event $[X^1_t=X^2_t]$ is not in $\sigma(Y_s,s\leqslant t)$.