# Characteristic function of time-changed Levy processes

Let $$X_t$$ be a Levy process, and $$Y_t$$ be a subordinator i.e. process with nondecreasing trajectories. I have to find characteristic function of $$X_{Y_t}$$. I know that I have to calculate: $$E[e^{iuX_{Y_t}}]=E[E[e^{iuX_k}|Y_t=k]]$$

as is written in "Time-changed Levyprocesses and option pricing" by Carr and Wu but I dont know how to calculate this

• I’m in a rush but don’t Carr and Wu have this JFE paper where they write the char fun of $X_{T_t}$ in terms of the Laplace transform of the random time $T_t$, evaluated at the char exponent of $X_t$ (or something along these lines?). They then give more details for different models. Do you have a particular model in mind? – Kevin Oct 17 '20 at 19:47
• @Kevin I would just like to see how to calculate this double expected value – HSmile Oct 18 '20 at 16:50