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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space, let $k, l \\ge 1$, and let $(\\xi_n)_{n \\ge 1}$ be a sequence of [[random variables|0PNRKMRC55]] with values in $\\R^l$.\nFor each $n \\ge 0$ let $f_n \\colon \\R^k \\times \\R^l \\to \\R^k$ be a Borel [[measurable function|YYNCJSR0PJ]].\nA *stochastic difference equation* is a recursion\n$$X_{n+1} = f_n(X_n, \\xi_{n+1}), \\qquad n \\ge 0,$$\ntogether with an initial random variable $X_0$ with values in $\\R^k$.\nIts *solution* is the [[stochastic process|0QH5ENKZM1]] $(X_n)_{n \\ge 0}$ defined recursively by these equations.\n\n## Remarks\n\nExample: 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 $f_n(x, e) = ax + e$.\nIts solution is $X_n = a^n X_0 + \\sum_{j=1}^{n} a^{n-j} \\varepsilon_j$.\n\nNon-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.\n\nHigher-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.\nIf $X_0, \\xi_1, \\xi_2, \\dots$ are independent, the solution is a Markov chain.\nIt is time-homogeneous when $f_n$ does not depend on $n$ and the $\\xi_n$ are identically distributed.\n",
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