We prove the existence of the persistence exponent $$\logλ:=\lim_{n\to\infty}\frac{1}{n}\log \mathbb{P}_μ(X_0\in S,\ldots,X_n\in S)$$ for a class of time homogeneous Markov chains $\{X_i\}_{i\geq 0}$ taking values in a Polish space, where $S$ is a Borel measurable set and $μ$ is an initial distribution. Focusing on the case of AR($p$) and MA($q$) processes with $p,q\in \mathbb{N}$ and continuous innovation distribution, we study the existence of $λ$ and its continuity in the parameters of the AR and MA processes, respectively, for $S=\mathbb{R}_{\geq 0}$. For AR processes with log-concave innovation distribution, we prove the strict monotonicity of $λ$. Finally, we compute new explicit exponents in several concrete examples.