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Show that q(x, t, y, s) satisfies the differential equation qt = (1/2)qxx. What equation does q(x, t, y, s) satisfy in the variables s and y (think about the relationship between qt and qs and q^ and qyy before you start computing)?

Keeping with the ordering of time in Figure 7.8, let us compute the covariance of W(t) and W(s):

E{W (t) W (s)} = E{( W (t) — W (s))W (s)} + E({ W (s)2})

=s where the last line of Eq. (7.29) follows because W(s) — W(0) and W(t) — W(s) are independent random variables, with mean 0. Suppose that we had interchanged the order of t and s. Our conclusion would then be that E{ W(t) W(s)} = t. In other words

and we are now ready to think about the derivative of Brownian motion.

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