# Random Walk

Jun 29th, 2015
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A random walk, RW(t), is the sum of the current and past observations of a white noise process, e(t) which we can assume for convenience has mean zero and variance one:

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I. Random WalkA random walk, RW(t), is the sum of the current and past observations of a white noise process, e(t) which we can assume for convenience has mean zero and variance one:RW(t) = e(t) + e(t-1) + e(t-2) + e(t-3) + ... .The mean function for this random walk is:E[RW(t)] = E e(t) + E e(t-1) + E e(t-2) + ... = 0.The variance or autocovariance at lag zero is:?RW,RW(0) = E[[e(t) + e(t-1) + e(t-2) ...]*[e(t) + e(t-1) + e(t-2) ...]} = ???+ ?????????????????????? ,noting that the expectation of cross-product terms such as E[e(t)*e(t-1)] are zero because of independence. Since the variance of a random walk is infinite it is clearly evolutionary. Although time series of prices often are approximated by a simple random walk model, prices are usually bounded below at zero and may practically be bounded above as well, so the random walk model is a simple abstraction of the actual behavior of prices. If the random walk series is lagged one period:RW(t-1) = e(t-1) + e(t-2) + e(t-3) + e(t-4) + ... , and subtracting RW(t-1) from RW(t), or differencing RW(t):RW(t) - RW(t-1) = ? RW(t) = e(t),i.e. the first difference of a random walk is white noise. Thus differencing this evolutionary time series yields a stationary time series. The difference operator or filter converts a random walk to white noise. This is illustrated schematically in the following diagram:Conversely, the inverse transform converts white noise into a random walk:?-1? RW(t) = RW(t

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