Stochastic difference equation

Remarks

Example: the autoregressive equation of order $1$, $X_{n+1} = a X_n + \varepsilon_{n+1}$ with $a \in \R$ and independent identically distributed $\varepsilon_n$, is a stochastic difference equation with $f_n(x, e) = ax + e$. Its solution is $X_n = a^n X_0 + \sum_{j=1}^{n} a^{n-j} \varepsilon_j$.

Non-example: $X_{n+1} = a X_n$ with a constant $X_0$ involves no random input, so it is a deterministic difference equation whose solution is a single sequence.

Higher-order equations, such as the autoregressive model of order $p$, $X_{n+1} = \sum_{i=1}^{p} a_i X_{n+1-i} + \varepsilon_{n+1}$, become first order on $\R^p$ by stacking $p$ consecutive values. If $X_0, \xi_1, \xi_2, \dots$ are independent, the solution is a Markov chain. It is time-homogeneous when $f_n$ does not depend on $n$ and the $\xi_n$ are identically distributed.