I have this problem to solve.
Is the process [$\xi_k$], $k \in N:=\{n1<n2<...\}$ Markovian, where $\xi_t, t \in \mathbb{R}$ is random Markov process. [y] is the integer part of y.
I don't know, how to start solving this.
I have such definition of Markov process:
Random process $\xi_t$ is a Markov process if $ P(AB|\mathcal{F}_{=t}) = P(A|\mathcal{F}_{=t})P(B|\mathcal{F}_{=t}) \: \forall t\in T,\:A\in\mathcal{F}_{\leq t},\:B\in\mathcal{F}_{\geq t}$.
Where $\mathcal{F}_{\geq t}$ is $\sigma$-algebra: $\mathcal{F}_{\geq t} := \sigma \{\xi_s,s\geq t \in T\}$
I don't know: the fact that we use $\xi_k$, where k from set of integers makes our process a discrete-time Markovian process? I think new step will be to rewrite our σ-algebras $F_{≥t}$ for new process $[\xi_k]$ Thank you!
Not necessarily.
Counter-example. Consider a process $\xi(t)$ that jumps: i) from $0.5$ to $0$ with probability $1$; ii) from $0$ to $1$ with probability $1$; and iii) from $1$ to $0.5$ with probability $1$.
The instants of jump are indexed by the elements of $N$.
Remark. $\left(\xi(t)\right)$ is a Markov process (you may consider its right-continuous version).
In this case, $$0=\mathbb{P}\left(\left[\xi_{k+1}\right]=1\left|\left[\xi_k\right]=0,\left[\xi_{k-1}\right]=1 \right.\right)\neq\mathbb{P}\left(\left[\xi_{k+1}\right]=1\left|\left[\xi_k\right]=0,\left[\xi_{k-1}\right]=0 \right.\right)=1.$$
Therefore, $\left[\xi_k\right]$ is not Markov.