Markov process and non-deterministic random variables

Question

How do I show the following: If $Z_1$ and $Z_2$ are non-deterministic random variables and we define the process $(X_t)_{t\geq 0}$ by $X_t = Z_1 \cos(t)+ Z_2 \sin(t)$. I want to show that this is not a Markov process. What could be an intuitive argument for this fact?

Answer 1

Maybe I'm wrong but if $\mathcal F_t=\sigma \{X_s\mid s\leq t\}$, then $X_t\in \mathcal F_s$ for all $s\leq t$. In particular $$\mathbb P\{X_t\in A\mid \mathcal F_s\}=\mathbb P\{X_t\in A\}=\mathbb P\{X_t\in A\mid X_s\},$$ so it looks to be a Markov process.

Answer 2

If $(X_t)$ is a Markov process we would have for $t>t_1>t_2$ :$$ P(X_t\in A\mid X_{t_1}=x_1,X_{t_2}=x_2) = P(X_t\in A\mid X_{t_1}=x_1) $$ But for example $t=\frac34\pi$, $t_1=\frac12\pi$, $t_2=0$ this is not true right? Now how do I get on from here?