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# Fourier Lévy formulae

Using the following trick: (29)

we obtain the equation to calculate the double integral using Lévy Area. Now, to measure this area, we can integrate: Then the Fourier transformation of the density of ILA conditional on ∆W1, ∆W2 is given by: and is explicitly known by  (it is also given in Lévy’s original paper ) as: where, given R2 = (∆W1)2 + (∆W2)2:  The probability density function (pdf) for X can be obtained exactly by inverting the Fourier transform fX (w): and then the cumulative distribution function (cdf) is: leading, via its inverse, to the sample rule: (30)

Note that the variance of X is: So far as we know, the pdf for Y cannot be written down in exact form, but for small ∆t, we have:  (31)

which is the Fourier transform of another normal distribution with density: Samples of Y can then be made in the usual way. (32)

So the double integral {24} can be approximated using the formulae: (33)

We explicitly know that the total variance of the Lévy Area is: So, although {33} is an approximation, we can see that we recover the exact total variance required. Prof. Klaus Schmitz

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