We now square G(t) and take its expectation, remembering that E(dWdW/} = 0 if i =j and equals dt if i = j, so that all the cross terms vanish when we take the expectation, and we see that

Var{G(t)} = limN!^ £ a2e23tj dtj = a2e23sds = a2 i 1

and from this can determine the properties of Y(t).

Using the results we have just derived, confirm that (i) Y(t) is normally distributed, (ii) E{ Y(t)} = e— 3'Y(0), and (iii) Var{ Y(t)} = [a2(1

Note that when t is very large Var{ Y(t)} ~ a2/23, which is a very interesting result because it tells us how fluctuations, measured by a, and dissipation, measured by 3, are connected to create the variance of Y(t). In physical systems, this is called the "fluctuation-dissipation" theorem and another piece of physical insight (the Maxwell-Boltzmann distribution) allows one to determine a for physical systems (see Connections). Given the result of Exercise 7.10, we can also immediately write down the transition density for the Ornstein-Uhlenbeck process q(x, t, y, s)dx defined to be the probability that x < Y(t) < x + dx given that Y(s) = y. It looks terribly frightening, but is simply a mathematical statement of the results of Exercise 7.10 in which Y(0) is replaced by Y(s) = y:

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