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## CHARACTERISTIC FUNCTION OF TIME-CHANGED LEVY PROCESSES

We are interested in the question of characteristic function of time-changed Levy process where the random time $T_t$ has a closed form Laplace transform.  Let $Y_t = X_{T_t}$ and $T_t$ is independent of $X_t$.  Then the characteristic function of $Y_t$ is the Laplace transform of $T_t$ evaluated at the characteristic exponent of $X_t$ (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):

$L[x^{\beta-1} E^\gamma_{\alpha,\beta}(ax^\alpha)] = s^{-\beta}(1-as^{-\alpha})^{-\gamma}$

If we considered time-changed Brownian motion with MLF random times:

$Y(t) = \theta T_t + \sigma W(T_t)$

we set $s= i u \theta + \sigma^2 u^2/2$ 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

$\frac{d}{ds} \frac{\phi'(s)}{\phi(s)} = \frac{ (2\alpha\beta+\beta)s^\alpha - (\alpha\beta a + \beta a)}{s - a s^{1-\alpha}}$

By Raible’s Theorem, this is the Fourier transform of the measure $|x|^2 K(x) dx$ where $K(x)$ is the Levy density of the time-changed process.