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so that

^fdWl fW (t + dt)-W (t)l , , n-*}=limd'!°n ( d }=0 (7-7)

and we conclude that the average value of dW/dt is 0. But look what happens with the variance:

i/dW\21 i(W(t + dt)-W(t))2l dt , nw h llmd'!0^ ( ( tf ()) ^ llmd'!0 & (7-8)

but we had better stop right here, because we know what is going to happen with the limit - it does not exist. In other words, although the sample paths of Brownian motion are continuous, they are not differ-entiable, at least in the sense that the variance of the derivative exists. Later in this chapter, in the section on white noise (see p. 261), we will make sense of the derivative of Brownian motion. For now, I want to introduce one more strange property associated with Brownian motion and then spend some time using it.

Suppose that we have a function f (t,W) which is known and well understood and can be differentiated to our hearts' content and for which we want to findf(t + dt, w + dW) when dt (and thus E{dW2})

is small and t and W(t) = w are specified. We Taylor expand in the usual manner, using a subscript to denote a derivative f (t + dt, w + dW) =f (t, w) + ft dt + fWd W

+ 2 {fit dt2 + 2fw dtd W + fww d W2} + o(dt2 ) + o(dtdW) + o(dW2 ) (7.9)

and now we ask ''what are the terms that are order dt on the right hand side of this expression?" Once again, this can only make sense in terms of an expectation, since f (t + dt, w + d W) will be a random variable. So let us take the expectation and use the properties of the increment of Brownian motion

Ef f (t + dt, w + dW)}= f (t, w)+ftdt + 2fwwdt + o(dt) (7.10)

so that the particular property of Brownian motion that E{dW2} = dt translates into a Taylor expansion in which first derivatives with respect to dt and first and second derivatives with respect to dW are the same order of dt. This is an example of Ito calculus, due to the mathematician K. Ito; see Connections for more details. We will now explore the implications of this observation.

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