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  "body": "\n## Summary\n\nA stochastic difference equation is a difference equation driven by random inputs, such as $X_{n+1} = f(X_n, \\xi_{n+1})$ with a sequence of random noise terms $\\xi_n$.\nIts solution is a [[stochastic process|0QH5ENKZM1]] in discrete time rather than a single sequence.\nThe [[autoregressive model|2JC96FMGA1]] of time series analysis and the [[random walk|04VZE3FZXF]] are the basic linear examples.\nWhen the noise terms are independent of each other and of the start, the solution is a [[Markov chain|0G48PYM96Q]].\n\n## Explanations\n\n### L1 human\n\nA deterministic recursion computes the next state from the present one by a fixed rule.\nA stochastic recursion also adds a fresh random shock at each step.\nFor example, $X_{n+1} = \\tfrac{1}{2} X_n + \\xi_{n+1}$ with independent coin-flip shocks $\\xi_n = \\pm 1$ keeps pulling the state toward $0$ while the shocks push it around.\nOne then studies averages, spreads and long-run distributions instead of exact values.\n",
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