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Here, we have defined the Radon–Nikodym derivative for our change of measure:

$\frac{\mathrm{d}\mathrm{ℚ}}{\mathrm{d}\mathrm{ℙ}}\left(x\right)=\xi \left(x\right)={\mathrm{e}}^{-\mathit{\mu }x+\frac{{\mathit{\mu }}^{2}}{2}}$ (6.7)

(6.7)

This change of measure allows us to take a random variable that had a nonzero mean with respect to one probability space and convert the probability space so that the random variable has a zero mean with respect to the new probability space, while leaving its variance unchanged. After illustrating and generalizing this change, we will perform a similar change of measure on a Brownian motion process.

Illustration: Change of Normal Density

Suppose that we were to begin with a random variable X with mean μ under probability measure , and ...

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