Stochastic difference equation

Summary

A 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 $\xi_n$. Its solution is a stochastic process in discrete time rather than a single sequence. The autoregressive model of time series analysis and the random walk are the basic linear examples. When the noise terms are independent of each other and of the start, the solution is a Markov chain.