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Now the derivative of this function will be discontinuous; since the derivative is 1 if t < s, and is 0 if t > s, there is a jump at t = s (Figure 7.9). We are going to deal with this problem by using the approach of generalized functions described in Chapter 2 (and in the course of this, learn more about Gaussians).

We will replace the derivative (S/St)x(t, s) by an approximation that is smooth but in the limit has the discontinuity. Define a family of functions

which we recognize as the tail of the cumulative distribution function for the Gaussian with mean 0 and variance 1/n. That is, the density is

We then set

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