Gaussian white noise

The derivative of Brownian motion, which we shall denote by £(t) = dW/dt, is often called Gaussian white noise. It should already be clear where Gaussian comes from; the origin of white will be understood at the end of this section, and the use of noise comes from engineers, who see fluctuations as noise, not as the element of variation that may lead to selection; Jaynes (2003) has a particularly nice discussion of this point. We have already shown the E{£(t)} = 0 and that problems arise when we try to compute E{£(t)2} in the usual way because of the variance of Brownian motion (recall the discussion around Eq. (7.8)). So, we are going to sneak up on this derivative by computing the covariance

Note that I have exchanged the order of differentiation and integration in Eq. (7.31); we will do this once more in this chapter. In general, one needs to be careful about doing such exchanges; both are okay here (if you want to know more about this question, consult a good book on advanced analysis). We know that E{ W(t) W(s)} = min(t, s). Let us think about this covariance as a function of t, when s is held fixed, as if it were just a parameter (Figure 7.9)

Figure 7.9. (a) The covariance function x(t,s) = E{W(t) W(s)} = min(t,s), thought of as a function of t with s as a parameter. (b) The derivative of the covariance function is either 1 or 0 with a discontinuity at t = s. (c) We approximate the derivative by a smooth function

Xn which in the limit has the discontinuity. (d) The approximate derivative is the tail of the cumulative Gaussian from t s.

s t

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