I have a discrete-time, stochastic sequence $X_n$ on $\mathbb{R}^d$ given by: $$X_{n+1} = f(X_n, M_n, W_n)$$ with $W_n\in\sigma(X_n, M_n)$ but $M_n\perp X_n$ and also $M_n$ are i.i.d. My question is if the sequence $\{X_n\}$ is a Markov Chain.
Intuition says yes because conditioning on $X_i$ for $i<n$ does not give extra information for $X_{n+1}$ once we condition on $X_n.$ If my intuition is correct, I would appreciate it if someone outline a rigorous proof of the Markovian property by identifying a kernel $P(\cdot, dx).$ For simplicity, all kinds of smoothness assumption can be made for $f:\mathbb{R}^d\times\mathbb{R}^d\times\mathbb{R}^d\to \mathbb{R}^d.$