We are interested in the question of characteristic function of time-changed Levy process where the random time has a closed form Laplace transform. Let and is independent of . Then the characteristic function of is the Laplace transform of evaluated at the characteristic exponent of (timechangeLevy_JFE2004) for Negarive Inverse Gaussian, Variance Gamma, and CGMY models provide examples of this technique (CGMY-SVLevy-2003).
We are interested in random time changes based on Mittag-Leffler distributions, so we consider the Laplace transform formula (Haubold-Mathai-Saxena-MLF2009):
If we considered time-changed Brownian motion with MLF random times:
we set into the Laplace transform formula. In order to calculate the Levy density from the characteristic function, we have to use Sebastian Raible’s Theorem (Corollary 2.4): Sebastian_Raible
By Raible’s Theorem, this is the Fourier transform of the measure where is the Levy density of the time-changed process.