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Its implementation requires only a shot noise decomposition of the Lévy measure of the process (explained later) without the need for the transition density, which often will not be known explicitly and is rather expensive to evaluate numer- ically.
All we need to do is find a shot noise decomposition of the corresponding Lévy measure.
Many of the existing methods for simulating Lévy processes, such as the inverse Lévy measure method of Khintchine (Ferguson and Klass 1972) and the standard way of simulating compound Poisson processes, are special cases of shot noise decomposition with particular choices for H(r, v) and v.
We now show the implementation of the foregoing Lévy shot noise decomposition for the relatively simple case of a gamma process with Lévy measure given by
The shot noise decomposition of the Lévy measure in The- orem 1 could be used for deriving the series representation of
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