# Multiplicative functionals for Markov process: discrete time

Let $X_t$ be a Markov process w.r.t. to its natural filtration $(\mathcal F_t)$ on the space of cadlag functions on $\mathbb R_{\geq 0}$ and $(Z_t)_{t\geq 0}$ be a non-negative martingale s.t. $\mathsf E_xZ_t = 1$ for all $x,t$. Let $\mathsf{\tilde{P}}$ be such that $$\mathsf{\tilde{P}}_x(A):=\mathsf E_x[1_A\cdot Z_t]$$ for any $A\in \mathcal F_t$, then the family $(\mathsf{\tilde{P}}_x)$ defines a time-homogeneous Markov process iff $Z_t$ is multiplicative functional, i.e. $$Z_{t+s} = Z_t\cdot(Z_s\circ \theta_t)$$ where $\theta_t$ is the shift operator.
I wonder which form can $Z$ take for a discrete time Markov process. The motivation is that for random walks $Y$ there is closed form for $Z$ given by $$Z_n = h(Y_1)\cdot h(Y_2)\cdot\dots\cdot h(Y_n)$$ where $\mathsf E h(Y_1) = 1$.