I have a question about the ARCH(1) process. Let $(\Omega, \mathcal F, P)$ be a probability space, let $(Z_t)_{t \in \mathbb Z}$ be a sequence of i.i.d. real-valued random variables with mean zero and variance one. A stochastic process $(X_t)_{t \in \mathbb Z}$ is an ARCH(1)-process if it is strictly stationary and if, for all $t$ and some process $(\sigma_t)$ with $\sigma_t > 0$ for every $t$, one has $X_t = \sigma_t Z_t$ and $\sigma_t^2 = \alpha_0 + \alpha_1 X_{t-1}^2$, where $\alpha_0 > 0$ and $\alpha_1 \ge 0$. Let $\mathcal F_t$ be the natural filtration of the process $(X_t)$, i.e. $\mathcal F_t = \sigma(X_s; s \le t)$.
I want to prove that $(X_t)$ has the Markov property, i.e. for each $B \in \mathcal B(\mathbb R)$ and for all $s, t \in\mathbb Z$ with $s < t$, one has $P[X_t \in B \mid \mathcal F_s] = P[X_t \in B \mid X_s]$. Is that possible?